B
    dd^                @   s  d Z ddlmZmZ ddlmZmZmZm	Z	m
Z
mZ ddlmZ ddlmZ ddlmZ ddlmZmZ ddlmZmZ dd	lmZmZ dd
lmZ ddlmZ ddl m!Z! ddl"m#Z#m$Z$ ddl%m&Z& ddl'm(Z( ddl)m*Z* ddl+m,Z, ddl-m.Z. ddl/m0Z0m1Z1m2Z2m3Z3 ddl4m5Z6m7Z8m9Z9 ddl:m;Z;m<Z<m=Z= ddl>m?Z? ddl@mAZA ddlBmCZC ddlDmEZE eAe.fddZFeAe.fddZGeAe.fddZHeAd d! ZId"d# ZJi ZKG d$d% d%e?eZLG d&d' d'e&e?eeMZNd(S ))zSparse polynomial rings.     )AnyDict)addmulltlegtge)reduce)GeneratorType)Expr)igcdoo)Symbolsymbols)CantSympifysympify)multinomial_coefficients)IPolys)construct_domain)dmp_to_dictdmp_from_dict)DomainElement)PolynomialRing)heugcd)MonomialOps)lex)CoercionFailedGeneratorsErrorExactQuotientFailedMultivariatePolynomialError)DomainOrderbuild_options)expr_from_dict_dict_reorder_parallel_dict_from_expr)DefaultPrinting)public)is_sequence)pollutec             C   s   t | ||}|f|j S )a  Construct a polynomial ring returning ``(ring, x_1, ..., x_n)``.

    Parameters
    ==========

    symbols : str
        Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
    domain : :class:`~.Domain` or coercible
    order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``

    Examples
    ========

    >>> from sympy.polys.rings import ring
    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.orderings import lex

    >>> R, x, y, z = ring("x,y,z", ZZ, lex)
    >>> R
    Polynomial ring in x, y, z over ZZ with lex order
    >>> x + y + z
    x + y + z
    >>> type(_)
    <class 'sympy.polys.rings.PolyElement'>

    )PolyRinggens)r   domainorder_ring r0   ^/work/yifan.wang/ringdown/master-ringdown-env/lib/python3.7/site-packages/sympy/polys/rings.pyring#   s    r2   c             C   s   t | ||}||jfS )a  Construct a polynomial ring returning ``(ring, (x_1, ..., x_n))``.

    Parameters
    ==========

    symbols : str
        Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
    domain : :class:`~.Domain` or coercible
    order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``

    Examples
    ========

    >>> from sympy.polys.rings import xring
    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.orderings import lex

    >>> R, (x, y, z) = xring("x,y,z", ZZ, lex)
    >>> R
    Polynomial ring in x, y, z over ZZ with lex order
    >>> x + y + z
    x + y + z
    >>> type(_)
    <class 'sympy.polys.rings.PolyElement'>

    )r+   r,   )r   r-   r.   r/   r0   r0   r1   xringB   s    r3   c             C   s(   t | ||}tdd |jD |j |S )a  Construct a polynomial ring and inject ``x_1, ..., x_n`` into the global namespace.

    Parameters
    ==========

    symbols : str
        Symbol/Expr or sequence of str, Symbol/Expr (non-empty)
    domain : :class:`~.Domain` or coercible
    order : :class:`~.MonomialOrder` or coercible, optional, defaults to ``lex``

    Examples
    ========

    >>> from sympy.polys.rings import vring
    >>> from sympy.polys.domains import ZZ
    >>> from sympy.polys.orderings import lex

    >>> vring("x,y,z", ZZ, lex)
    Polynomial ring in x, y, z over ZZ with lex order
    >>> x + y + z # noqa:
    x + y + z
    >>> type(_)
    <class 'sympy.polys.rings.PolyElement'>

    c             S   s   g | ]
}|j qS r0   )name).0symr0   r0   r1   
<listcomp>}   s    zvring.<locals>.<listcomp>)r+   r*   r   r,   )r   r-   r.   r/   r0   r0   r1   vringa   s    r8   c       
         s   d}t | s| gd } }ttt| } t||}t| |\}}|jdkrtdd |D g }t||d\|_}t	t
||  fdd|D }t|j|j|j}tt|j|}	|r||	d fS ||	fS dS )	ad  Construct a ring deriving generators and domain from options and input expressions.

    Parameters
    ==========

    exprs : :class:`~.Expr` or sequence of :class:`~.Expr` (sympifiable)
    symbols : sequence of :class:`~.Symbol`/:class:`~.Expr`
    options : keyword arguments understood by :class:`~.Options`

    Examples
    ========

    >>> from sympy import sring, symbols

    >>> x, y, z = symbols("x,y,z")
    >>> R, f = sring(x + 2*y + 3*z)
    >>> R
    Polynomial ring in x, y, z over ZZ with lex order
    >>> f
    x + 2*y + 3*z
    >>> type(_)
    <class 'sympy.polys.rings.PolyElement'>

    FTNc             S   s   g | ]}t | qS r0   )listvalues)r5   repr0   r0   r1   r7      s    zsring.<locals>.<listcomp>)optc                s"   g | ]} fd d|  D qS )c                s   i | ]\}} | |qS r0   r0   )r5   mc)	coeff_mapr0   r1   
<dictcomp>   s    z$sring.<locals>.<listcomp>.<dictcomp>)items)r5   r;   )r?   r0   r1   r7      s    r   )r)   r9   mapr   r#   r&   r-   sumr   dictzipr+   r,   r.   	from_dict)
exprsr   optionsZsingler<   ZrepscoeffsZ
coeffs_domr/   polysr0   )r?   r1   sring   s     

rK   c             C   sr   t | tr| rt| ddS dS t | tr.| fS t| rftdd | D rPt| S tdd | D rf| S tdd S )NT)seqr0   c             s   s   | ]}t |tV  qd S )N)
isinstancestr)r5   sr0   r0   r1   	<genexpr>   s    z!_parse_symbols.<locals>.<genexpr>c             s   s   | ]}t |tV  qd S )N)rM   r   )r5   rO   r0   r0   r1   rP      s    zbexpected a string, Symbol or expression or a non-empty sequence of strings, Symbols or expressions)rM   rN   _symbolsr   r)   allr   )r   r0   r0   r1   _parse_symbols   s    

rS   c               @   s8  e Zd ZdZefddZdd Zdd Zdd	 Zd
d Z	dd Z
dd ZdEddZdd Zedd Zedd ZdFddZdd Zdd Zdd  ZeZdGd!d"ZdHd#d$Zd%d& Zd'd( Zd)d* Zd+d, Zd-d. Zd/d0 Zd1d2 Zd3d4 Zd5d6 Z ed7d8 Z!ed9d: Z"d;d< Z#d=d> Z$d?d@ Z%dAdB Z&dCdD Z'dS )Ir+   z*Multivariate distributed polynomial ring. c                s  t t|}t|}t|}t  | j||| f}t|}|d kr|j	rlt
|t
|j@ rltdt| }||_t||_tdtfd|i|_||_||_||_ |_d| |_| |_t
|j|_|j|jfg|_|r8t|}| |_ |! |_"|# |_$|% |_&|' |_(|) |_*|+ |_,n6dd }||_ ||_"dd |_$||_&||_(||_*||_, t-krt.|_/n fdd|_/xFt0|j|jD ]4\}	}
t1|	t2r|	j3}t4||st5|||
 qW |t|< |S )	Nz7polynomial ring and it's ground domain share generatorsPolyElementr2   )r   c             S   s   dS )Nr0   r0   )abr0   r0   r1   <lambda>       z"PolyRing.__new__.<locals>.<lambda>c             S   s   dS )Nr0   r0   )rU   rV   r>   r0   r0   r1   rW      rX   c                s   t |  dS )N)key)max)f)r.   r0   r1   rW      rX   )6tuplerS   len	DomainOpt
preprocessOrderOpt__name___ring_cachegetis_Compositesetr   r   object__new___hash_tuplehash_hashtyperT   dtypengensr-   r.   
zero_monom_gensr,   	_gens_setone_oner   r   monomial_mulpowmonomial_powZmulpowmonomial_mulpowZldivmonomial_ldivdivmonomial_divlcmZmonomial_lcmgcdmonomial_gcdr   rZ   leading_expvrE   rM   r   r4   hasattrsetattr)clsr   r-   r.   rm   rh   objZcodegenZmonunitsymbol	generatorr4   r0   )r.   r1   rg      s`    















zPolyRing.__new__c             C   sJ   | j j}g }x4t| jD ]&}| |}| j}|||< || qW t|S )z(Return a list of polynomial generators. )r-   rq   rangerm   monomial_basiszeroappendr\   )selfrq   ro   iexpvpolyr0   r0   r1   ro     s    
zPolyRing._gensc             C   s   | j | j| jfS )N)r   r-   r.   )r   r0   r0   r1   __getnewargs__  s    zPolyRing.__getnewargs__c             C   s:   | j  }|d= x$| D ]\}}|dr||= qW |S )Nr}   Z	monomial_)__dict__copyrA   
startswith)r   staterY   valuer0   r0   r1   __getstate__  s    


zPolyRing.__getstate__c             C   s   | j S )N)rj   )r   r0   r0   r1   __hash__   s    zPolyRing.__hash__c             C   s2   t |to0| j| j| j| jf|j|j|j|jfkS )N)rM   r+   r   r-   rm   r.   )r   otherr0   r0   r1   __eq__#  s    
zPolyRing.__eq__c             C   s
   | |k S )Nr0   )r   r   r0   r0   r1   __ne__(  s    zPolyRing.__ne__Nc             C   s    |  |p| j|p| j|p| jS )N)	__class__r   r-   r.   )r   r   r-   r.   r0   r0   r1   clone+  s    zPolyRing.clonec             C   s   dg| j  }d||< t|S )zReturn the ith-basis element. r      )rm   r\   )r   r   Zbasisr0   r0   r1   r   .  s    zPolyRing.monomial_basisc             C   s   |   S )N)rl   )r   r0   r0   r1   r   4  s    zPolyRing.zeroc             C   s   |  | jS )N)rl   rr   )r   r0   r0   r1   rq   8  s    zPolyRing.onec             C   s   | j ||S )N)r-   convert)r   elementorig_domainr0   r0   r1   
domain_new<  s    zPolyRing.domain_newc             C   s   |  | j|S )N)term_newrn   )r   coeffr0   r0   r1   
ground_new?  s    zPolyRing.ground_newc             C   s    |  |}| j}|r|||< |S )N)r   r   )r   monomr   r   r0   r0   r1   r   B  s
    
zPolyRing.term_newc             C   s   t |trF| |jkr|S t | jtr<| jj|jkr<| |S tdnxt |trZtdndt |trn| 	|S t |t
ry
| |S  tk
r   | |S X nt |tr| |S | |S d S )N
conversionZparsing)rM   rT   r2   r-   r   r   NotImplementedErrorrN   rD   rF   r9   
from_terms
ValueError	from_listr   	from_expr)r   r   r0   r0   r1   ring_newI  s$    











zPolyRing.ring_newc             C   s<   | j }| j}x*| D ]\}}|||}|r|||< qW |S )N)r   r   rA   )r   r   r   r   r   r   r   r0   r0   r1   rF   a  s    
zPolyRing.from_dictc             C   s   |  t||S )N)rF   rD   )r   r   r   r0   r0   r1   r   l  s    zPolyRing.from_termsc             C   s   |  t|| jd | jS )Nr   )rF   r   rm   r-   )r   r   r0   r0   r1   r   o  s    zPolyRing.from_listc                s$   j  fdd  t|S )Nc                s    | }|d k	r|S | jr2tttt | jS | jrNtttt | jS | 	 \}}|j
rx|dkrx |t| S | S d S )Nr   )rc   Zis_Addr
   r   r9   rB   argsZis_Mulr   Zas_base_expZ
is_Integerintr   r   )exprr   baseexp)_rebuildr-   mappingr   r0   r1   r   u  s    
z(PolyRing._rebuild_expr.<locals>._rebuild)r-   r   )r   r   r   r0   )r   r-   r   r   r1   _rebuild_exprr  s    zPolyRing._rebuild_exprc             C   sZ   t tt| j| j}y| ||}W n$ tk
rJ   td| |f Y nX | |S d S )Nz@expected an expression convertible to a polynomial in %s, got %s)	rD   r9   rE   r   r,   r   r   r   r   )r   r   r   r   r0   r0   r1   r     s    zPolyRing.from_exprc             C   s   |dkr| j rd}qd}nt|trj|}d|kr<|| j k r<q| j  |kr\|dkr\| d }qtd| nt|| jry| j|}W q tk
r   td| Y qX nJt|try| j|}W q tk
r   td| Y qX ntd| |S )z+Compute index of ``gen`` in ``self.gens``. Nr   r   zinvalid generator index: %szinvalid generator: %szEexpected a polynomial generator, an integer, a string or None, got %s)	rm   rM   r   r   rl   r,   indexrN   r   )r   genr   r0   r0   r1   r     s.    

zPolyRing.indexc                sB   t t| j|  fddt| jD }|s2| jS | j|dS dS )z,Remove specified generators from this ring. c                s   g | ]\}}| kr|qS r0   r0   )r5   r   rO   )indicesr0   r1   r7     s    z!PolyRing.drop.<locals>.<listcomp>)r   N)re   rB   r   	enumerater   r-   r   )r   r,   r   r0   )r   r1   drop  s
    zPolyRing.dropc             C   s$   | j | }|s| jS | j|dS d S )N)r   )r   r-   r   )r   rY   r   r0   r0   r1   __getitem__  s    
zPolyRing.__getitem__c             C   s6   | j jst| j dr$| j| j j dS td| j  d S )Nr-   )r-   z%s is not a composite domain)r-   rd   r~   r   r   )r   r0   r0   r1   	to_ground  s    zPolyRing.to_groundc             C   s   t | S )N)r   )r   r0   r0   r1   	to_domain  s    zPolyRing.to_domainc             C   s   ddl m} || j| j| jS )Nr   )	FracField)Zsympy.polys.fieldsr   r   r-   r.   )r   r   r0   r0   r1   to_field  s    zPolyRing.to_fieldc             C   s   t | jdkS )Nr   )r]   r,   )r   r0   r0   r1   is_univariate  s    zPolyRing.is_univariatec             C   s   t | jdkS )Nr   )r]   r,   )r   r0   r0   r1   is_multivariate  s    zPolyRing.is_multivariatec             G   s<   | j }x0|D ](}t|tdr,|| j| 7 }q||7 }qW |S )aw  
        Add a sequence of polynomials or containers of polynomials.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> R, x = ring("x", ZZ)
        >>> R.add([ x**2 + 2*i + 3 for i in range(4) ])
        4*x**2 + 24
        >>> _.factor_list()
        (4, [(x**2 + 6, 1)])

        )include)r   r)   r   r   )r   objspr   r0   r0   r1   r     s    
zPolyRing.addc             G   s<   | j }x0|D ](}t|tdr,|| j| 9 }q||9 }qW |S )a  
        Multiply a sequence of polynomials or containers of polynomials.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> R, x = ring("x", ZZ)
        >>> R.mul([ x**2 + 2*i + 3 for i in range(4) ])
        x**8 + 24*x**6 + 206*x**4 + 744*x**2 + 945
        >>> _.factor_list()
        (1, [(x**2 + 3, 1), (x**2 + 5, 1), (x**2 + 7, 1), (x**2 + 9, 1)])

        )r   )rq   r)   r   r   )r   r   r   r   r0   r0   r1   r     s    
zPolyRing.mulc                s`   t t| j|  fddt| jD } fddt| jD }|sH| S | j|| j| dS dS )zd
        Remove specified generators from the ring and inject them into
        its domain.
        c                s   g | ]\}}| kr|qS r0   r0   )r5   r   rO   )r   r0   r1   r7     s    z+PolyRing.drop_to_ground.<locals>.<listcomp>c                s   g | ]\}}| kr|qS r0   r0   )r5   r   r   )r   r0   r1   r7     s    )r   r-   N)re   rB   r   r   r   r,   r   r   )r   r,   r   r0   )r   r1   drop_to_ground  s    zPolyRing.drop_to_groundc             C   s6   | |kr.t | jt |j}| jt|dS | S dS )z+Add the generators of ``other`` to ``self``)r   N)re   r   unionr   r9   )r   r   symsr0   r0   r1   compose  s    zPolyRing.composec             C   s$   t | jt |}| jt|dS )z9Add the elements of ``symbols`` as generators to ``self``)r   )re   r   r   r   r9   )r   r   r   r0   r0   r1   add_gens&  s    zPolyRing.add_gens)NNN)N)N)N)(ra   
__module____qualname____doc__r   rg   ro   r   r   r   r   r   r   r   propertyr   rq   r   r   r   r   __call__rF   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r0   r0   r0   r1   r+      sF   A






r+   c               @   s"  e Zd ZdZdd Zdd Zdd ZdZd	d
 Zdd Z	dd Z
dd Zdd Zdd Zdd Zdd Zdd ZdddZdd Zdd  Zd!d" Zd#d$ Zd%d& Zd'd( Zd)d* Zd+d, Zd-d. Zd/d0 Zd1d2 Zd3d4 Zd5d6 Zed7d8 Z ed9d: Z!ed;d< Z"ed=d> Z#ed?d@ Z$edAdB Z%edCdD Z&edEdF Z'edGdH Z(edIdJ Z)edKdL Z*edMdN Z+edOdP Z,edQdR Z-edSdT Z.edUdV Z/edWdX Z0dYdZ Z1d[d\ Z2d]d^ Z3d_d` Z4dadb Z5dcdd Z6dedf Z7dgdh Z8didj Z9dkdl Z:dmdn Z;dodp Z<dqdr Z=dsdt Z>dudv Z?dwdx Z@dydz ZAd{d| ZBeAZCeBZDd}d~ ZEdd ZFdd ZGdd ZHdd ZIdd ZJdd ZKdddZLdd ZMdddZNdd ZOdd ZPdd ZQdd ZRdd ZSedd ZTedd ZUdd ZVedd ZWdd ZXdd ZYdddZZdddZ[dddZ\dd Z]dd Z^dd Z_dd Z`dd Zadd Zbdd Zcdd Zddd Zedd Zfdd ZgddĄ ZhddƄ ZiddȄ Zjddʄ Zkdd̄ ZlelZmdd΄ ZnddЄ Zodd҄ ZpddԄ Zqddք Zrdd؄ Zsddڄ Ztdd܄ Zuddބ Zvdd Zwdd Zxdd Zydd Zzdd Z{dd Z|dd Z}dd Z~dd Zd ddZd!ddZd"ddZdd Zdd Zdd Zdd Zdd  Zdd Zdd Zdd Zdd Zd	d
 Zdd Zdd Zdd Zdd Zdd Zd#ddZdd ZdS ($  rT   z5Element of multivariate distributed polynomial ring. c             C   s
   |  |S )N)r   )r   initr0   r0   r1   new/  s    zPolyElement.newc             C   s
   | j  S )N)r2   r   )r   r0   r0   r1   parent2  s    zPolyElement.parentc             C   s   | j t|  fS )N)r2   r9   	iterterms)r   r0   r0   r1   r   5  s    zPolyElement.__getnewargs__Nc             C   s.   | j }|d kr*t| jt|  f | _ }|S )N)rj   ri   r2   	frozensetrA   )r   rj   r0   r0   r1   r   :  s    zPolyElement.__hash__c             C   s
   |  | S )a  Return a copy of polynomial self.

        Polynomials are mutable; if one is interested in preserving
        a polynomial, and one plans to use inplace operations, one
        can copy the polynomial. This method makes a shallow copy.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> R, x, y = ring('x, y', ZZ)
        >>> p = (x + y)**2
        >>> p1 = p.copy()
        >>> p2 = p
        >>> p[R.zero_monom] = 3
        >>> p
        x**2 + 2*x*y + y**2 + 3
        >>> p1
        x**2 + 2*x*y + y**2
        >>> p2
        x**2 + 2*x*y + y**2 + 3

        )r   )r   r0   r0   r1   r   E  s    zPolyElement.copyc             C   sZ   | j |kr| S | j j|jkrFttt| | j j|j }||| j jS || | j jS d S )N)r2   r   r9   rE   r%   r   r-   rF   )r   new_ringtermsr0   r0   r1   set_ringa  s    
zPolyElement.set_ringc             G   sH   |r.t || jjkr.td| jjt |f n| jj}t|  f| S )Nz&not enough symbols, expected %s got %s)r]   r2   rm   r   r   r$   as_expr_dict)r   r   r0   r0   r1   as_exprj  s    zPolyElement.as_exprc                s    | j jj  fdd|  D S )Nc                s   i | ]\}} ||qS r0   r0   )r5   r   r   )to_sympyr0   r1   r@   t  s    z,PolyElement.as_expr_dict.<locals>.<dictcomp>)r2   r-   r   r   )r   r0   )r   r1   r   r  s    
zPolyElement.as_expr_dictc                s|   | j j}|jr|js|j| fS | }|j |j}|j}x|  D ]}| || qBW | 	 fdd| 
 D } |fS )Nc                s   g | ]\}}||  fqS r0   r0   )r5   kv)commonr0   r1   r7     s    z,PolyElement.clear_denoms.<locals>.<listcomp>)r2   r-   is_Fieldhas_assoc_Ringrq   get_ringrz   denomr:   r   rA   )r   r-   Zground_ringrz   r   r   r   r0   )r   r1   clear_denomsv  s    
zPolyElement.clear_denomsc             C   s(   x"t |  D ]\}}|s| |= qW dS )z+Eliminate monomials with zero coefficient. N)r9   rA   )r   r   r   r0   r0   r1   
strip_zero  s    zPolyElement.strip_zeroc             C   sR   |s
|  S t |tr,|j| jkr,t| |S t| dkr<dS | | jj|kS dS )aP  Equality test for polynomials.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> p1 = (x + y)**2 + (x - y)**2
        >>> p1 == 4*x*y
        False
        >>> p1 == 2*(x**2 + y**2)
        True

        r   FN)rM   rT   r2   rD   r   r]   rc   rn   )p1p2r0   r0   r1   r     s    zPolyElement.__eq__c             C   s
   | |k S )Nr0   )r   r   r0   r0   r1   r     s    zPolyElement.__ne__c             C   s   | j }t||jrdt|  t| kr.dS |jj}x(|  D ]}|| | || |s@dS q@W dS t| dkrtdS y|j|}W n t	k
r   dS X |j| 
 ||S dS )z+Approximate equality test for polynomials. FTr   N)r2   rM   rl   re   keysr-   almosteqr]   r   r   const)r   r   Z	tolerancer2   r   r   r0   r0   r1   r     s     zPolyElement.almosteqc             C   s   t | |  fS )N)r]   r   )r   r0   r0   r1   sort_key  s    zPolyElement.sort_keyc             C   s(   t || jjr ||  | S tS d S )N)rM   r2   rl   r   NotImplemented)r   r   opr0   r0   r1   _cmp  s    zPolyElement._cmpc             C   s   |  |tS )N)r   r   )r   r   r0   r0   r1   __lt__  s    zPolyElement.__lt__c             C   s   |  |tS )N)r   r   )r   r   r0   r0   r1   __le__  s    zPolyElement.__le__c             C   s   |  |tS )N)r   r   )r   r   r0   r0   r1   __gt__  s    zPolyElement.__gt__c             C   s   |  |tS )N)r   r	   )r   r   r0   r0   r1   __ge__  s    zPolyElement.__ge__c             C   sH   | j }||}|jdkr$||jfS t|j}||= ||j|dfS d S )Nr   )r   )r2   r   rm   r-   r9   r   r   )r   r   r2   r   r   r0   r0   r1   _drop  s    



zPolyElement._dropc             C   s   |  |\}}| jjdkr8| jr*| dS td| nT|j}xH|  D ]<\}}|| dkrxt|}||= ||t	|< qHtd| qHW |S d S )Nr   zCannot drop %sr   )
r   r2   rm   	is_groundr   r   r   rA   r9   r\   )r   r   r   r2   r   r   r   Kr0   r0   r1   r     s    
zPolyElement.dropc             C   s6   | j }||}t|j}||= ||j||| dfS )N)r   r-   )r2   r   r9   r   r   )r   r   r2   r   r   r0   r0   r1   _drop_to_ground  s
    

zPolyElement._drop_to_groundc             C   s   | j jdkrtd| |\}}|j}|jjd }xn|  D ]b\}}|d | ||d d   }||kr|||  |||< q>||  |||  |7  < q>W |S )Nr   z$Cannot drop only generator to groundr   )	r2   rm   r   r   r   r-   r,   r   
mul_ground)r   r   r   r2   r   r   r   monr0   r0   r1   r     s    "zPolyElement.drop_to_groundc             C   s   t | | jjd | jjS )Nr   )r   r2   rm   r-   )r   r0   r0   r1   to_dense  s    zPolyElement.to_densec             C   s   t | S )N)rD   )r   r0   r0   r1   to_dict  s    zPolyElement.to_dictc             C   s  | s| | jjjS |d }|d }| j}|j}|j}	|j}
g }xD|  D ]6\}}|j|}|rjdnd}|	| ||
kr| |}|r|
dr|dd  }n,|r| }|| jjkr|j||dd}nd	}g }xt|	D ]}|| }|sq|j|| |dd}|dkrR|t|ks(|d
k r:|j||dd}n|}|	|||f  q|	d|  qW |rt|g| }|	|| qLW |d
 dkr|d
}|dkr|d
d d	|S )NZMulZAtomz - z + -r   T)strict r   Fz%s)z + z - )Z_printr2   r-   r   r   rm   rn   r   is_negativer   r   rq   Zparenthesizer   r   joinpopinsert)r   printer
precedenceZexp_patternZ
mul_symbolZprec_mulZ	prec_atomr2   r   rm   zmZsexpvsr   r   negativesignZscoeffZsexpvr   r   r   Zsexpheadr0   r0   r1   rN     sT    





zPolyElement.strc             C   s   | | j jkS )N)r2   rp   )r   r0   r0   r1   is_generatorC  s    zPolyElement.is_generatorc             C   s   |  pt | dko| jj| kS )Nr   )r]   r2   rn   )r   r0   r0   r1   r   G  s    zPolyElement.is_groundc             C   s   |  pt | dko| jdkS )Nr   )r]   LC)r   r0   r0   r1   is_monomialK  s    zPolyElement.is_monomialc             C   s   t | dkS )Nr   )r]   )r   r0   r0   r1   is_termO  s    zPolyElement.is_termc             C   s   | j j| jS )N)r2   r-   r   r  )r   r0   r0   r1   r   S  s    zPolyElement.is_negativec             C   s   | j j| jS )N)r2   r-   is_positiver  )r   r0   r0   r1   r  W  s    zPolyElement.is_positivec             C   s   | j j| jS )N)r2   r-   is_nonnegativer  )r   r0   r0   r1   r  [  s    zPolyElement.is_nonnegativec             C   s   | j j| jS )N)r2   r-   is_nonpositiver  )r   r0   r0   r1   r  _  s    zPolyElement.is_nonpositivec             C   s   |  S )Nr0   )r[   r0   r0   r1   is_zeroc  s    zPolyElement.is_zeroc             C   s   | | j jkS )N)r2   rq   )r[   r0   r0   r1   is_oneg  s    zPolyElement.is_onec             C   s   | j j| jS )N)r2   r-   r	  r  )r[   r0   r0   r1   is_monick  s    zPolyElement.is_monicc             C   s   | j j|  S )N)r2   r-   r	  content)r[   r0   r0   r1   is_primitiveo  s    zPolyElement.is_primitivec             C   s   t dd |  D S )Nc             s   s   | ]}t |d kV  qdS )r   N)rC   )r5   r   r0   r0   r1   rP   u  s    z(PolyElement.is_linear.<locals>.<genexpr>)rR   
itermonoms)r[   r0   r0   r1   	is_linears  s    zPolyElement.is_linearc             C   s   t dd |  D S )Nc             s   s   | ]}t |d kV  qdS )   N)rC   )r5   r   r0   r0   r1   rP   y  s    z+PolyElement.is_quadratic.<locals>.<genexpr>)rR   r  )r[   r0   r0   r1   is_quadraticw  s    zPolyElement.is_quadraticc             C   s   | j jsdS | j | S )NT)r2   rm   Z	dmp_sqf_p)r[   r0   r0   r1   is_squarefree{  s    zPolyElement.is_squarefreec             C   s   | j jsdS | j | S )NT)r2   rm   Zdmp_irreducible_p)r[   r0   r0   r1   is_irreducible  s    zPolyElement.is_irreduciblec             C   s    | j jr| j | S tdd S )Nzcyclotomic polynomial)r2   r   Zdup_cyclotomic_pr    )r[   r0   r0   r1   is_cyclotomic  s    zPolyElement.is_cyclotomicc             C   s   |  dd |  D S )Nc             S   s   g | ]\}}|| fqS r0   r0   )r5   r   r   r0   r0   r1   r7     s    z'PolyElement.__neg__.<locals>.<listcomp>)r   r   )r   r0   r0   r1   __neg__  s    zPolyElement.__neg__c             C   s   | S )Nr0   )r   r0   r0   r1   __pos__  s    zPolyElement.__pos__c       
      C   sB  |s|   S | j}t||jrp|   }|j}|jj}x6| D ]*\}}|||| }|rb|||< q>||= q>W |S t|trt|jt	r|jj|jkrn*t|jjt	r|jjj|kr|
| S tS y||}W n tk
r   tS X |   }|s|S |j}	|	|  kr|||	< n(|||	  kr*||	= n||	  |7  < |S dS )a  Add two polynomials.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> (x + y)**2 + (x - y)**2
        2*x**2 + 2*y**2

        N)r   r2   rM   rl   rc   r-   r   rA   rT   r   __radd__r   r   r   rn   r   )
r   r   r2   r   rc   r   r   r   Zcp2r   r0   r0   r1   __add__  sB    




zPolyElement.__add__c             C   s   |   }|s|S | j}y||}W n tk
r8   tS X |j}||  krV|||< n&|||  krl||= n||  |7  < |S d S )N)r   r2   r   r   r   rn   r   )r   nr   r2   r   r0   r0   r1   r    s    
zPolyElement.__radd__c       	      C   s:  |s|   S | j}t||jrp|   }|j}|jj}x6| D ]*\}}|||| }|rb|||< q>||= q>W |S t|trt|jt	r|jj|jkrn*t|jjt	r|jjj|kr|
| S tS y||}W n tk
r   tS X |   }|j}||  kr| ||< n&||| kr"||= n||  |8  < |S dS )a.  Subtract polynomial p2 from p1.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> p1 = x + y**2
        >>> p2 = x*y + y**2
        >>> p1 - p2
        -x*y + x

        N)r   r2   rM   rl   rc   r-   r   rA   rT   r   __rsub__r   r   r   rn   r   )	r   r   r2   r   rc   r   r   r   r   r0   r0   r1   __sub__  s>    



zPolyElement.__sub__c             C   s\   | j }y||}W n tk
r(   tS X |j}x| D ]}| |  ||< q6W ||7 }|S dS )a#  n - p1 with n convertible to the coefficient domain.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x + y
        >>> 4 - p
        -x - y + 4

        N)r2   r   r   r   r   )r   r  r2   r   r   r0   r0   r1   r    s    
zPolyElement.__rsub__c             C   sD  | j }|j}| r|s|S t||jr|j}|jj}|j}t| }xF|  D ]:\}}	x0|D ](\}
}|||
}||||	|  ||< q\W qNW |	  |S t|t
rt|jtr|jj |j krn*t|j jtr|j jj |kr|| S tS y||}W n tk
r   tS X x,|  D ] \}}	|	| }|r|||< qW |S dS )a!  Multiply two polynomials.

        Examples
        ========

        >>> from sympy.polys.domains import QQ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', QQ)
        >>> p1 = x + y
        >>> p2 = x - y
        >>> p1*p2
        x**2 - y**2

        N)r2   r   rM   rl   rc   r-   rs   r9   rA   r   rT   r   __rmul__r   r   r   )r   r   r2   r   rc   r   rs   Zp2itexp1v1Zexp2Zv2r   r   r0   r0   r1   __mul__/  s<    


zPolyElement.__mul__c             C   sh   | j j}|s|S y|j |}W n tk
r4   tS X x(|  D ]\}}|| }|r@|||< q@W |S dS )a  p2 * p1 with p2 in the coefficient domain of p1.

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x + y
        >>> 4 * p
        4*x + 4*y

        N)r2   r   r   r   r   rA   )r   r   r   r  r  r   r0   r0   r1   r  a  s    zPolyElement.__rmul__c             C   s   | j }|s| r|jS tdnXt| dkrvt|  d \}}|j}|dkr^|||||< n|| ||||< |S t|}|dk rtdnT|dkr| 	 S |dkr| 
 S |dkr| | 
  S t| dkr| |S | |S dS )	a(  raise polynomial to power `n`

        Examples
        ========

        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.rings import ring

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x + y**2
        >>> p**3
        x**3 + 3*x**2*y**2 + 3*x*y**4 + y**6

        z0**0r   r   zNegative exponentr        N)r2   rq   r   r]   r9   rA   r   ru   r   r   square_pow_multinomial_pow_generic)r   r  r2   r   r   r   r0   r0   r1   __pow__~  s0    


zPolyElement.__pow__c             C   sD   | j j}| }x2|d@ r,|| }|d8 }|s,P | }|d }qW |S )Nr   r  )r2   rq   r!  )r   r  r   r>   r0   r0   r1   r#    s    zPolyElement._pow_genericc             C   s   t t| | }| jj}| jj}|  }| jjj}| jj}x|D ]\}}	|}
|	}x6t||D ](\}\}}|r^||
||}
||| 9 }q^W t	|
}|}|
||| }|r|||< qB||krB||= qBW |S )N)r   r]   rA   r2   rv   rn   r-   r   rE   r\   rc   )r   r  Zmultinomialsrv   rn   r   r   r   ZmultinomialZmultinomial_coeffZproduct_monomZproduct_coeffr   r   r   r0   r0   r1   r"    s*    


zPolyElement._pow_multinomialc             C   s   | j }|j}|j}t|  }|jj}|j}xbtt|D ]R}|| }| | }	x<t|D ]0}
||
 }|||}||||	| |   ||< qXW q:W |	d}|j}x4| 
 D ](\}}|||}||||d  ||< qW |  |S )a  square of a polynomial

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x + y**2
        >>> p.square()
        x**2 + 2*x*y**2 + y**4

        r  )r2   r   rc   r9   r   r-   rs   r   r]   imul_numrA   r   )r   r2   r   rc   r   r   rs   r   Zk1pkjZk2r   r   r   r0   r0   r1   r!    s(    
"

zPolyElement.squarec             C   s   | j }|stdnft||jr*| |S t|trzt|jtrP|jj |j krPn*t|j jtrv|j jj |krv|| S t	S y|
|}W n tk
r   t	S X | || |fS d S )Nzpolynomial division)r2   ZeroDivisionErrorrM   rl   rx   rT   r-   r   __rdivmod__r   r   r   
quo_ground
rem_ground)r   r   r2   r0   r0   r1   
__divmod__  s     



zPolyElement.__divmod__c             C   s   t S )N)r   )r   r   r0   r0   r1   r)    s    zPolyElement.__rdivmod__c             C   s   | j }|stdnft||jr*| |S t|trzt|jtrP|jj |j krPn*t|j jtrv|j jj |krv|| S t	S y|
|}W n tk
r   t	S X | |S d S )Nzpolynomial division)r2   r(  rM   rl   remrT   r-   r   __rmod__r   r   r   r+  )r   r   r2   r0   r0   r1   __mod__  s     



zPolyElement.__mod__c             C   s   t S )N)r   )r   r   r0   r0   r1   r.  .  s    zPolyElement.__rmod__c             C   s   | j }|stdnzt||jr>|jr2| |d  S | |S nPt|trt|jtrd|jj |j krdn*t|j jtr|j jj |kr|	| S t
S y||}W n tk
r   t
S X | |S d S )Nzpolynomial divisionr   )r2   r(  rM   rl   r  quorT   r-   r   __rtruediv__r   r   r   r*  )r   r   r2   r0   r0   r1   __truediv__1  s$    


zPolyElement.__truediv__c             C   s   t S )N)r   )r   r   r0   r0   r1   r1  J  s    zPolyElement.__rtruediv__c                sJ   | j j| j j}|j | j j|jr6 fdd}n fdd}|S )Nc                sF   | \}}|\}}|kr|}n
||}|d k	r>| ||fS d S d S )Nr0   )	a_lm_a_lc	b_lm_b_lca_lma_lcb_lmb_lcr   )
domain_quory   r   r0   r1   term_divY  s    
z'PolyElement._term_div.<locals>.term_divc                sN   | \}}|\}}|kr|}n
||}|d ksF|| sF| ||fS d S d S )Nr0   )r3  r4  r5  r6  r7  r8  r   )r9  ry   r   r0   r1   r:  e  s    
)r2   rn   r-   r0  ry   r   )r   r-   r:  r0   )r9  ry   r   r1   	_term_divR  s    zPolyElement._term_divc                s  | j  d}t|trd}|g}t|s.td| sL|rB j jfS g  jfS x|D ]}|j  krRtdqRW t|} fddt|D }| 	 } j}| 
 }dd |D }	x|rxd}
d}x|
|k rP|dkrP| }|||| f|	|
 ||
 |	|
  f}|d	k	rF|\}}||
 ||f||
< |||
 || f}d
}q|
d
7 }
qW |s| }|||| f}||= qW | jkr||7 }|r|s j|fS |d |fS n||fS d	S )aU  Division algorithm, see [CLO] p64.

        fv array of polynomials
           return qv, r such that
           self = sum(fv[i]*qv[i]) + r

        All polynomials are required not to be Laurent polynomials.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> f = x**3
        >>> f0 = x - y**2
        >>> f1 = x - y
        >>> qv, r = f.div((f0, f1))
        >>> qv[0]
        x**2 + x*y**2 + y**4
        >>> qv[1]
        0
        >>> r
        y**6

        FTzpolynomial divisionz"self and f must have the same ringc                s   g | ]
} j qS r0   )r   )r5   r   )r2   r0   r1   r7     s    z#PolyElement.div.<locals>.<listcomp>c             S   s   g | ]}|  qS r0   )r}   )r5   Zfxr0   r0   r1   r7     s    r   Nr   )r2   rM   rT   rR   r(  r   r   r]   r   r   r;  r}   _iadd_monom_iadd_poly_monomrn   )r   ZfvZ
ret_singler[   rO   Zqvr   rr:  Zexpvsr   Zdivoccurredr   termZexpv1r>   r0   )r2   r1   rx   s  sV    



&


zPolyElement.divc             C   sJ  | }t |tr|g}t|s$td|j}|j}|j}|j}|j}| }|j	}	|
 }|j}
x|rDx|D ]}||	|j	}|d k	rl|\}}xD| D ]8\}}|||}|
||||  }|s||= q|||< qW | }|d k	r||| f}	P qlW |	\}}||kr||  |7  < n|||< ||= | }|d k	r`||| f}	q`W |S )Nzpolynomial division)rM   rT   rR   r(  r2   r-   r   rs   r;  LTr   rc   r   r}   )r   Gr[   r2   r-   r   rs   r>  r:  Zltfrc   gZtqr=   r>   mgcgm1c1ZltmZltcr0   r0   r1   r-    sL    



zPolyElement.remc             C   s   |  |d S )Nr   )rx   )r[   rA  r0   r0   r1   r0    s    zPolyElement.quoc             C   s$   |  |\}}|s|S t| |d S )N)rx   r   )r[   rA  qr>  r0   r0   r1   exquo  s    zPolyElement.exquoc             C   s^   | | j jkr|  }n| }|\}}||}|dkr>|||< n||7 }|rT|||< n||= |S )a  add to self the monomial coeff*x0**i0*x1**i1*...
        unless self is a generator -- then just return the sum of the two.

        mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...)

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x**4 + 2*y
        >>> m = (1, 2)
        >>> p1 = p._iadd_monom((m, 5))
        >>> p1
        x**4 + 5*x*y**2 + 2*y
        >>> p1 is p
        True
        >>> p = x
        >>> p1 = p._iadd_monom((m, 5))
        >>> p1
        5*x*y**2 + x
        >>> p1 is p
        False

        N)r2   rp   r   rc   )r   mcZcpselfr   r   r>   r0   r0   r1   r<    s    



zPolyElement._iadd_monomc             C   s   | }||j jkr| }|\}}|j}|j jj}|j j}xD| D ]8\}	}
||	|}||||
|  }|rt|||< qB||= qBW |S )aE  add to self the product of (p)*(coeff*x0**i0*x1**i1*...)
        unless self is a generator -- then just return the sum of the two.

        mc is a tuple, (monom, coeff), where monomial is (i0, i1, ...)

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y, z = ring('x, y, z', ZZ)
        >>> p1 = x**4 + 2*y
        >>> p2 = y + z
        >>> m = (1, 2, 3)
        >>> p1 = p1._iadd_poly_monom(p2, (m, 3))
        >>> p1
        x**4 + 3*x*y**3*z**3 + 3*x*y**2*z**4 + 2*y

        )r2   rp   r   rc   r-   r   rs   rA   )r   r   rI  r   r=   r>   rc   r   rs   r   r   kar   r0   r0   r1   r=  #  s    



zPolyElement._iadd_poly_monomc                s@   | j | | st S  dk r"dS t fdd|  D S dS )z
        The leading degree in ``x`` or the main variable.

        Note that the degree of 0 is negative infinity (the SymPy object -oo).

        r   c                s   g | ]}|  qS r0   r0   )r5   r   )r   r0   r1   r7   V  s    z&PolyElement.degree.<locals>.<listcomp>N)r2   r   r   rZ   r  )r[   xr0   )r   r1   degreeH  s    zPolyElement.degreec             C   s2   | st  f| jj S ttttt|   S dS )z
        A tuple containing leading degrees in all variables.

        Note that the degree of 0 is negative infinity (the SymPy object -oo)

        N)	r   r2   rm   r\   rB   rZ   r9   rE   r  )r[   r0   r0   r1   degreesX  s    zPolyElement.degreesc                s@   | j | | st S  dk r"dS t fdd|  D S dS )z
        The tail degree in ``x`` or the main variable.

        Note that the degree of 0 is negative infinity (the SymPy object -oo)

        r   c                s   g | ]}|  qS r0   r0   )r5   r   )r   r0   r1   r7   r  s    z+PolyElement.tail_degree.<locals>.<listcomp>N)r2   r   r   minr  )r[   rK  r0   )r   r1   tail_degreed  s    zPolyElement.tail_degreec             C   s2   | st  f| jj S ttttt|   S dS )z
        A tuple containing tail degrees in all variables.

        Note that the degree of 0 is negative infinity (the SymPy object -oo)

        N)	r   r2   rm   r\   rB   rN  r9   rE   r  )r[   r0   r0   r1   tail_degreest  s    zPolyElement.tail_degreesc             C   s   | r| j | S dS dS )aT  Leading monomial tuple according to the monomial ordering.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y, z = ring('x, y, z', ZZ)
        >>> p = x**4 + x**3*y + x**2*z**2 + z**7
        >>> p.leading_expv()
        (4, 0, 0)

        N)r2   r}   )r   r0   r0   r1   r}     s    zPolyElement.leading_expvc             C   s   |  || jjjS )N)rc   r2   r-   r   )r   r   r0   r0   r1   
_get_coeff  s    zPolyElement._get_coeffc             C   sp   |dkr|  | jjS t|| jjr`t| }t|dkr`|d \}}|| jjj	kr`|  |S t
d| dS )a  
        Returns the coefficient that stands next to the given monomial.

        Parameters
        ==========

        element : PolyElement (with ``is_monomial = True``) or 1

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y, z = ring("x,y,z", ZZ)
        >>> f = 3*x**2*y - x*y*z + 7*z**3 + 23

        >>> f.coeff(x**2*y)
        3
        >>> f.coeff(x*y)
        0
        >>> f.coeff(1)
        23

        r   r   zexpected a monomial, got %sN)rQ  r2   rn   rM   rl   r9   r   r]   r-   rq   r   )r   r   r   r   r   r0   r0   r1   r     s    
zPolyElement.coeffc             C   s   |  | jjS )z!Returns the constant coeffcient. )rQ  r2   rn   )r   r0   r0   r1   r     s    zPolyElement.constc             C   s   |  |  S )N)rQ  r}   )r   r0   r0   r1   r    s    zPolyElement.LCc             C   s    |   }|d kr| jjS |S d S )N)r}   r2   rn   )r   r   r0   r0   r1   LM  s    zPolyElement.LMc             C   s&   | j j}|  }|r"| j jj||< |S )a  
        Leading monomial as a polynomial element.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> (3*x*y + y**2).leading_monom()
        x*y

        )r2   r   r}   r-   rq   )r   r   r   r0   r0   r1   leading_monom  s
    zPolyElement.leading_monomc             C   s4   |   }|d kr"| jj| jjjfS || |fS d S )N)r}   r2   rn   r-   r   rQ  )r   r   r0   r0   r1   r@    s    zPolyElement.LTc             C   s(   | j j}|  }|dk	r$| | ||< |S )a  Leading term as a polynomial element.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> (3*x*y + y**2).leading_term()
        3*x*y

        N)r2   r   r}   )r   r   r   r0   r0   r1   leading_term  s
    zPolyElement.leading_termc                sP    d kr| j j n
t   tkr6t|dd ddS t| fddddS d S )Nc             S   s   | d S )Nr   r0   )r   r0   r0   r1   rW     rX   z%PolyElement._sorted.<locals>.<lambda>T)rY   reversec                s    | d S )Nr   r0   )r   )r.   r0   r1   rW     rX   )r2   r.   r`   r_   r   sorted)r   rL   r.   r0   )r.   r1   _sorted  s    

zPolyElement._sortedc             C   s   dd |  |D S )a  Ordered list of polynomial coefficients.

        Parameters
        ==========

        order : :class:`~.MonomialOrder` or coercible, optional

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.orderings import lex, grlex

        >>> _, x, y = ring("x, y", ZZ, lex)
        >>> f = x*y**7 + 2*x**2*y**3

        >>> f.coeffs()
        [2, 1]
        >>> f.coeffs(grlex)
        [1, 2]

        c             S   s   g | ]\}}|qS r0   r0   )r5   _r   r0   r0   r1   r7      s    z&PolyElement.coeffs.<locals>.<listcomp>)r   )r   r.   r0   r0   r1   rI     s    zPolyElement.coeffsc             C   s   dd |  |D S )a
  Ordered list of polynomial monomials.

        Parameters
        ==========

        order : :class:`~.MonomialOrder` or coercible, optional

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.orderings import lex, grlex

        >>> _, x, y = ring("x, y", ZZ, lex)
        >>> f = x*y**7 + 2*x**2*y**3

        >>> f.monoms()
        [(2, 3), (1, 7)]
        >>> f.monoms(grlex)
        [(1, 7), (2, 3)]

        c             S   s   g | ]\}}|qS r0   r0   )r5   r   rX  r0   r0   r1   r7   :  s    z&PolyElement.monoms.<locals>.<listcomp>)r   )r   r.   r0   r0   r1   monoms"  s    zPolyElement.monomsc             C   s   |  t|  |S )a  Ordered list of polynomial terms.

        Parameters
        ==========

        order : :class:`~.MonomialOrder` or coercible, optional

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ
        >>> from sympy.polys.orderings import lex, grlex

        >>> _, x, y = ring("x, y", ZZ, lex)
        >>> f = x*y**7 + 2*x**2*y**3

        >>> f.terms()
        [((2, 3), 2), ((1, 7), 1)]
        >>> f.terms(grlex)
        [((1, 7), 1), ((2, 3), 2)]

        )rW  r9   rA   )r   r.   r0   r0   r1   r   <  s    zPolyElement.termsc             C   s   t |  S )z,Iterator over coefficients of a polynomial. )iterr:   )r   r0   r0   r1   
itercoeffsV  s    zPolyElement.itercoeffsc             C   s   t |  S )z)Iterator over monomials of a polynomial. )rZ  r   )r   r0   r0   r1   r  Z  s    zPolyElement.itermonomsc             C   s   t |  S )z%Iterator over terms of a polynomial. )rZ  rA   )r   r0   r0   r1   r   ^  s    zPolyElement.itertermsc             C   s   t |  S )z+Unordered list of polynomial coefficients. )r9   r:   )r   r0   r0   r1   
listcoeffsb  s    zPolyElement.listcoeffsc             C   s   t |  S )z(Unordered list of polynomial monomials. )r9   r   )r   r0   r0   r1   
listmonomsf  s    zPolyElement.listmonomsc             C   s   t |  S )z$Unordered list of polynomial terms. )r9   rA   )r   r0   r0   r1   	listtermsj  s    zPolyElement.listtermsc             C   sF   | | j jkr| | S |s$|   dS x| D ]}| |  |9  < q*W | S )a:  multiply inplace the polynomial p by an element in the
        coefficient ring, provided p is not one of the generators;
        else multiply not inplace

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring('x, y', ZZ)
        >>> p = x + y**2
        >>> p1 = p.imul_num(3)
        >>> p1
        3*x + 3*y**2
        >>> p1 is p
        True
        >>> p = x
        >>> p1 = p.imul_num(3)
        >>> p1
        3*x
        >>> p1 is p
        False

        N)r2   rp   clear)r   r>   r   r0   r0   r1   r%  n  s    
zPolyElement.imul_numc             C   s4   | j j}|j}|j}x|  D ]}|||}qW |S )z*Returns GCD of polynomial's coefficients. )r2   r-   r   r{   r[  )r[   r-   contr{   r   r0   r0   r1   r    s    zPolyElement.contentc             C   s   |   }|| |fS )z,Returns content and a primitive polynomial. )r  r*  )r[   r`  r0   r0   r1   	primitive  s    zPolyElement.primitivec             C   s   | s| S |  | jS dS )z5Divides all coefficients by the leading coefficient. N)r*  r  )r[   r0   r0   r1   monic  s    zPolyElement.monicc                s,    s| j jS  fdd|  D }| |S )Nc                s   g | ]\}}||  fqS r0   r0   )r5   r   r   )rK  r0   r1   r7     s    z*PolyElement.mul_ground.<locals>.<listcomp>)r2   r   r   r   )r[   rK  r   r0   )rK  r1   r     s    zPolyElement.mul_groundc                s*   | j j fdd|  D }| |S )Nc                s   g | ]\}}| |fqS r0   r0   )r5   f_monomf_coeff)r   rs   r0   r1   r7     s    z)PolyElement.mul_monom.<locals>.<listcomp>)r2   rs   rA   r   )r[   r   r   r0   )r   rs   r1   	mul_monom  s    zPolyElement.mul_monomc                sZ   |\ | r s| j jS | j jkr.|  S | j j fdd|  D }| |S )Nc                s"   g | ]\}}||  fqS r0   r0   )r5   rc  rd  )r   r   rs   r0   r1   r7     s    z(PolyElement.mul_term.<locals>.<listcomp>)r2   r   rn   r   rs   rA   r   )r[   r?  r   r0   )r   r   rs   r1   mul_term  s    
zPolyElement.mul_termc                sl   | j j}std| r"|jkr&| S |jrL|j  fdd|  D }nfdd|  D }| |S )Nzpolynomial divisionc                s   g | ]\}}| |fqS r0   r0   )r5   r   r   )r0  rK  r0   r1   r7     s    z*PolyElement.quo_ground.<locals>.<listcomp>c                s$   g | ]\}}|  s||  fqS r0   r0   )r5   r   r   )rK  r0   r1   r7     s    )r2   r-   r(  rq   r   r0  r   r   )r[   rK  r-   r   r0   )r0  rK  r1   r*    s    zPolyElement.quo_groundc                sl    \}}|st dn"| s"| jjS || jjkr8| |S |   fdd|  D }| dd |D S )Nzpolynomial divisionc                s   g | ]}| qS r0   r0   )r5   t)r?  r:  r0   r1   r7     s    z(PolyElement.quo_term.<locals>.<listcomp>c             S   s   g | ]}|d k	r|qS )Nr0   )r5   rg  r0   r0   r1   r7     s    )r(  r2   r   rn   r*  r;  r   r   )r[   r?  r   r   r   r0   )r?  r:  r1   quo_term  s    

zPolyElement.quo_termc                s|   | j jjrPg }xV|  D ]2\}}|  }| d kr<|  }|||f qW n fdd|  D }| |}|  |S )Nr  c                s   g | ]\}}||  fqS r0   r0   )r5   r   r   )r   r0   r1   r7     s    z,PolyElement.trunc_ground.<locals>.<listcomp>)r2   r-   is_ZZr   r   r   r   )r[   r   r   r   r   r   r0   )r   r1   trunc_ground  s    

zPolyElement.trunc_groundc             C   sB   | }|  }|  }|jj||}||}||}|||fS )N)r  r2   r-   r{   r*  )r   rB  r[   fcgcr{   r0   r0   r1   extract_ground  s    

zPolyElement.extract_groundc                s6   | s| j jjS | j jj | fdd|  D S d S )Nc                s   g | ]} |qS r0   r0   )r5   r   )
ground_absr0   r1   r7     s    z%PolyElement._norm.<locals>.<listcomp>)r2   r-   r   absr[  )r[   Z	norm_funcr0   )rn  r1   _norm  s    

zPolyElement._normc             C   s
   |  tS )N)rp  rZ   )r[   r0   r0   r1   max_norm  s    zPolyElement.max_normc             C   s
   |  tS )N)rp  rC   )r[   r0   r0   r1   l1_norm	  s    zPolyElement.l1_normc             G   s  | j }| gt| }dg|j }xF|D ]>}x8| D ],}x&t|D ]\}}t|| |||< qBW q4W q&W x t|D ]\}}	|	srd||< qrW t|}tdd |D r||fS g }
xR|D ]J}|j}x4|	 D ](\}}dd t
||D }||t|< qW |
| qW ||
fS )Nr   r   c             s   s   | ]}|d kV  qdS )r   Nr0   )r5   rV   r0   r0   r1   rP     s    z&PolyElement.deflate.<locals>.<genexpr>c             S   s   g | ]\}}|| qS r0   r0   )r5   r   r'  r0   r0   r1   r7   &  s    z'PolyElement.deflate.<locals>.<listcomp>)r2   r9   rm   r  r   r   r\   rR   r   r   rE   r   )r[   rA  r2   rJ   Jr   r   r   r=   rV   HhIr   Nr0   r0   r1   deflate  s*    

zPolyElement.deflatec             C   sB   | j j}x4|  D ](\}}dd t||D }||t|< qW |S )Nc             S   s   g | ]\}}|| qS r0   r0   )r5   r   r'  r0   r0   r1   r7   1  s    z'PolyElement.inflate.<locals>.<listcomp>)r2   r   r   rE   r\   )r[   rs  r   rv  r   rw  r0   r0   r1   inflate-  s
    zPolyElement.inflatec             C   sf   | }|j j}|js6| \}}| \}}|||}|| ||}|jsZ||S | S d S )N)	r2   r-   r   ra  rz   r0  r{   r   rb  )r   rB  r[   r-   rk  rl  r>   ru  r0   r0   r1   rz   6  s    
zPolyElement.lcmc             C   s   |  |d S )Nr   )	cofactors)r[   rB  r0   r0   r1   r{   F  s    zPolyElement.gcdc             C   s   | s|s| j j}|||fS | s8| |\}}}|||fS |sV|| \}}}|||fS t| dkr|| |\}}}|||fS t|dkr|| \}}}|||fS | |\}\} }| |\}}}||||||fS )Nr   )r2   r   	_gcd_zeror]   
_gcd_monomrx  _gcdry  )r[   rB  r   ru  cffcfgrs  r0   r0   r1   rz  I  s$    




zPolyElement.cofactorsc             C   s4   | j j| j j }}|jr"|||fS | || fS d S )N)r2   rq   r   r  )r[   rB  rq   r   r0   r0   r1   r{  _  s    
zPolyElement._gcd_zeroc                s   | j }|jj}|jj|j}|jt|  d \}}||  x(| D ]\}}||| | qJW |  fg}	| || fg}
|  fdd| D }|	|
|fS )Nr   c                s$   g | ]\}}|| fqS r0   r0   )r5   rC  rD  )_cgcd_mgcd
ground_quorw   r0   r1   r7   s  s    z*PolyElement._gcd_monom.<locals>.<listcomp>)	r2   r-   r{   r0  r|   rw   r9   r   r   )r[   rB  r2   Z
ground_gcdr|   mfcfrC  rD  ru  r~  r  r0   )r  r  r  rw   r1   r|  f  s    

"zPolyElement._gcd_monomc             C   s:   | j }|jjr| |S |jjr*| |S || |S d S )N)r2   r-   Zis_QQ_gcd_QQri  _gcd_ZZZdmp_inner_gcd)r[   rB  r2   r0   r0   r1   r}  v  s    

zPolyElement._gcdc             C   s
   t | |S )N)r   )r[   rB  r0   r0   r1   r    s    zPolyElement._gcd_ZZc             C   s   | }|j }|j|j d}| \}}| \}}||}||}||\}}}	||}|j|  }
}||	|j
|
|}|	|	|j
|
|}	|||	fS )N)r-   )r2   r   r-   r   r   r   r  r  rb  r   r0  )r   rB  r[   r2   r   r  rD  ru  r~  r  r>   r0   r0   r1   r    s    


zPolyElement._gcd_QQc             C   s  | }|j }|s||jfS |j}|jr*|js<||\}}}n|j| d}| \}	}| \}
}|	|}|	|}||\}}}|j|
|	\}}
}	|	|}|	|}|
|
}|
|	}| }||jkr|| }}n2||j kr | |  }}n|
|}|
|}||fS )a  
        Cancel common factors in a rational function ``f/g``.

        Examples
        ========

        >>> from sympy.polys import ring, ZZ
        >>> R, x,y = ring("x,y", ZZ)

        >>> (2*x**2 - 2).cancel(x**2 - 2*x + 1)
        (2*x + 2, x - 1)

        )r-   )r2   rq   r-   r   r   rz  r   r   r   r   r   canonical_unit)r   rB  r[   r2   r-   rX  r   rG  r   Zcqcpur0   r0   r1   cancel  s4    









zPolyElement.cancelc             C   s   | j j}|| jS )N)r2   r-   r  r  )r[   r-   r0   r0   r1   r    s    zPolyElement.canonical_unitc       	      C   sd   | j }||}||}|j}x>|  D ]2\}}|| r*|||}||||  ||< q*W |S )a!  Computes partial derivative in ``x``.

        Examples
        ========

        >>> from sympy.polys.rings import ring
        >>> from sympy.polys.domains import ZZ

        >>> _, x, y = ring("x,y", ZZ)
        >>> p = x + x**2*y**3
        >>> p.diff(x)
        2*x*y**3 + 1

        )r2   r   r   r   r   rw   r   )	r[   rK  r2   r   r=   rB  r   r   er0   r0   r1   diff  s    

zPolyElement.diffc             G   sT   dt |  k r| jjkr8n n| tt| jj|S td| jjt |f d S )Nr   z1expected at least 1 and at most %s values, got %s)r]   r2   rm   evaluater9   rE   r,   r   )r[   r:   r0   r0   r1   r     s     zPolyElement.__call__c                sP  | }t |tr`|d kr`|d |dd   \ }}| |}|sD|S  fdd|D }||S |j}||}|j|}|jdkr|jj}x&|	 D ]\\}}||||  7 }qW |S |
|j}	x|	 D ]t\}
}|
| |
d | |
|d d    }}
|||  }|
|	kr8||	|
  }|r0||	|
< n|	|
= q|r||	|
< qW |	S d S )Nr   r   c                s   g | ]\}}|  |fqS r0   )r   )r5   YrU   )Xr0   r1   r7     s    z(PolyElement.evaluate.<locals>.<listcomp>)rM   r9   r  r2   r   r-   r   rm   r   r   r   )r   rK  rU   r[   r2   r   resultr  r   r   r   r0   )r  r1   r    s8    


&

zPolyElement.evaluatec             C   s,  | }t |tr8|d kr8x|D ]\}}|||}qW |S |j}||}|j|}|jdkr|jj}x&|	 D ]\\}}	||	||  7 }qpW |
|S |j}
x|	 D ]x\}}	|| |d | d ||d d    }}|	||  }	||
kr|	|
|  }	|	r|	|
|< n|
|= q|	r|	|
|< qW |
S d S )Nr   )r   )rM   r9   subsr2   r   r-   r   rm   r   r   r   )r   rK  rU   r[   r  r2   r   r  r  r   r   r   r0   r0   r1   r  	  s2    


*

zPolyElement.subsc                s(  | j }|j}ttt|jtt|j |d k	r>||fg}nDt|trRt|}n0t|trzt	t|
  fddd}ntdx.t|D ]"\}\}} | ||f||< qW xp|  D ]d\}}	t|}|j}
x2|D ]*\}}|| d }||< |r|
|| 9 }
qW |
t||	f}
||
7 }qW |S )Nc                s    | d  S )Nr   r0   )r   )gens_mapr0   r1   rW   Q	  rX   z%PolyElement.compose.<locals>.<lambda>)rY   z9expected a generator, value pair a sequence of such pairsr   )r2   r   rD   r9   rE   r,   r   rm   rM   rV  rA   r   r   r   r   rq   rf  r\   )r[   rK  rU   r2   r   Zreplacementsr   rB  r   r   Zsubpolyr   r  r0   )r  r1   r   F	  s,    


zPolyElement.composec             C   s   | j | |S )N)r2   Zdmp_pdiv)r[   rB  r0   r0   r1   pdivi	  s    zPolyElement.pdivc             C   s   | j | |S )N)r2   Zdmp_prem)r[   rB  r0   r0   r1   preml	  s    zPolyElement.premc             C   s   | j | |S )N)r2   Zdmp_quo)r[   rB  r0   r0   r1   pquoo	  s    zPolyElement.pquoc             C   s   | j | |S )N)r2   Z	dmp_exquo)r[   rB  r0   r0   r1   pexquor	  s    zPolyElement.pexquoc             C   s   | j | |S )N)r2   Zdmp_half_gcdex)r[   rB  r0   r0   r1   
half_gcdexu	  s    zPolyElement.half_gcdexc             C   s   | j | |S )N)r2   Z	dmp_gcdex)r[   rB  r0   r0   r1   gcdexx	  s    zPolyElement.gcdexc             C   s   | j | |S )N)r2   Zdmp_subresultants)r[   rB  r0   r0   r1   subresultants{	  s    zPolyElement.subresultantsc             C   s   | j | |S )N)r2   Zdmp_resultant)r[   rB  r0   r0   r1   	resultant~	  s    zPolyElement.resultantc             C   s   | j | S )N)r2   Zdmp_discriminant)r[   r0   r0   r1   discriminant	  s    zPolyElement.discriminantc             C   s    | j jr| j | S tdd S )Nzpolynomial decomposition)r2   r   Zdup_decomposer    )r[   r0   r0   r1   	decompose	  s    zPolyElement.decomposec             C   s"   | j jr| j | |S tdd S )Nzpolynomial shift)r2   r   Z	dup_shiftr    )r[   rU   r0   r0   r1   shift	  s    zPolyElement.shiftc             C   s    | j jr| j | S tdd S )Nzsturm sequence)r2   r   Z	dup_sturmr    )r[   r0   r0   r1   sturm	  s    zPolyElement.sturmc             C   s   | j | S )N)r2   Zdmp_gff_list)r[   r0   r0   r1   gff_list	  s    zPolyElement.gff_listc             C   s   | j | S )N)r2   Zdmp_sqf_norm)r[   r0   r0   r1   sqf_norm	  s    zPolyElement.sqf_normc             C   s   | j | S )N)r2   Zdmp_sqf_part)r[   r0   r0   r1   sqf_part	  s    zPolyElement.sqf_partFc             C   s   | j j| |dS )N)rR   )r2   Zdmp_sqf_list)r[   rR   r0   r0   r1   sqf_list	  s    zPolyElement.sqf_listc             C   s   | j | S )N)r2   Zdmp_factor_list)r[   r0   r0   r1   factor_list	  s    zPolyElement.factor_list)N)N)N)N)N)N)N)N)N)F)ra   r   r   r   r   r   r   rj   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   rN   r   r  r   r  r  r   r  r  r  r  r	  r
  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r$  r#  r"  r!  r,  r)  r/  r.  r2  r1  __floordiv____rfloordiv__r;  rx   r-  r0  rH  r<  r=  rL  rM  rO  rP  r}   rQ  r   r   r  rR  rS  r@  rT  rW  rI   rY  r   r[  r  r   r\  r]  r^  r%  r  ra  rb  r   re  rf  r*  rh  rj  r+  rm  rp  rq  rr  rx  ry  rz   r{   rz  r{  r|  r}  r  r  r  r  r  r   r  r  r   r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r  r0   r0   r0   r1   rT   ,  s   	06620$!L-,%%#!	
8,'#
rT   N)Or   typingr   r   ZtDictoperatorr   r   r   r   r   r	   	functoolsr
   typesr   Zsympy.core.exprr   Zsympy.core.numbersr   r   Zsympy.core.symbolr   r   rQ   Zsympy.core.sympifyr   r   Zsympy.ntheory.multinomialr   Zsympy.polys.compatibilityr   Zsympy.polys.constructorr   Zsympy.polys.densebasicr   r   Z!sympy.polys.domains.domainelementr   Z"sympy.polys.domains.polynomialringr   Zsympy.polys.heuristicgcdr   Zsympy.polys.monomialsr   Zsympy.polys.orderingsr   Zsympy.polys.polyerrorsr   r   r   r    Zsympy.polys.polyoptionsr!   r^   r"   r`   r#   Zsympy.polys.polyutilsr$   r%   r&   Zsympy.printing.defaultsr'   Zsympy.utilitiesr(   Zsympy.utilities.iterablesr)   Zsympy.utilities.magicr*   r2   r3   r8   rK   rS   rb   r+   rD   rT   r0   r0   r0   r1   <module>   sJ    5  j