a d?Z-d@dAZ.dBdCZ/dDdEZ0dFdGZ1dHdIZ2dS)L)Dummy) nextprimecrt)PolynomialRing)gf_gcd gf_from_dictgf_gcdexgf_divgf_lcm)ModularGCDFailed)sqrtNcCs|j}|s|s|j|j|jfS|sR|j|jjkrB| |j|j fS||j|jfSn2|s|j|jjkrv| |j |jfS||j|jfSdS)zn Compute the GCD of two polynomials in trivial cases, i.e. when one or both polynomials are zero. N)ringzeroLCdomainone)fgrrf/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/polys/modulargcd.py _trivial_gcd srcCs|jj}|rj|}|}||j|}|}||kr8q`||||f||j|}q&|}|}q|||j||S)zM Compute the GCD of two univariate polynomials in `\mathbb{Z}_p[x]`. )rrdegreeinvertr mul_monom mul_ground trunc_ground)fpgppdomremdeglcinvdegremrrr_gf_gcd#s(r%cCsb|jj|j|j}d}t|}||dkr6t|}q ||}||}t|||}|}|S)a Compute an upper bound for the degree of the GCD of two univariate integer polynomials `f` and `g`. The function chooses a suitable prime `p` and computes the GCD of `f` and `g` in `\mathbb{Z}_p[x]`. The choice of `p` guarantees that the degree in `\mathbb{Z}_p[x]` is greater than or equal to the degree in `\mathbb{Z}[x]`. Parameters ========== f : PolyElement univariate integer polynomial g : PolyElement univariate integer polynomial r)rrgcdrrrr%r)rrgammarrrhpdeghprrr_degree_bound_univariate:s     r+cCsn|}|jjd}|jj}t|dD]8}t||g||||||gddd||f<q(||S)a Construct a polynomial `h_{pq}` in `\mathbb{Z}_{p q}[x]` such that .. math :: h_{pq} = h_p \; \mathrm{mod} \, p h_{pq} = h_q \; \mathrm{mod} \, q for relatively prime integers `p` and `q` and polynomials `h_p` and `h_q` in `\mathbb{Z}_p[x]` and `\mathbb{Z}_q[x]` respectively. The coefficients of the polynomial `h_{pq}` are computed with the Chinese Remainder Theorem. The symmetric representation in `\mathbb{Z}_p[x]`, `\mathbb{Z}_q[x]` and `\mathbb{Z}_{p q}[x]` is used. It is assumed that `h_p` and `h_q` have the same degree. Parameters ========== hp : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_p` hq : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_q` p : Integer modulus of `h_p`, relatively prime to `q` q : Integer modulus of `h_q`, relatively prime to `p` Examples ======== >>> from sympy.polys.modulargcd import _chinese_remainder_reconstruction_univariate >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> p = 3 >>> q = 5 >>> hp = -x**3 - 1 >>> hq = 2*x**3 - 2*x**2 + x >>> hpq = _chinese_remainder_reconstruction_univariate(hp, hq, p, q) >>> hpq 2*x**3 + 3*x**2 + 6*x + 5 >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True rr&TZ symmetric)rrgensrrangercoeffZ strip_zero)r)hqrqnxhpqirrr,_chinese_remainder_reconstruction_univariate[s6 6r6cCs|j|jkr|jjjsJt||}|dur0|S|j}|\}}|\}}|j||}t||}|dkr||||||||fS|j|j|j}d} d} t | } || dkrt | } q| | } | | } t | | | } | }||krqn||krd} |}q| | | } | dkr4| } | }qt | || | }| | 9} ||ksZ|}q||}||\}}||\}}|s|s|jdkr| }||}|||}|||}|||fSqdS)a Computes the GCD of two polynomials in `\mathbb{Z}[x]` using a modular algorithm. The algorithm computes the GCD of two univariate integer polynomials `f` and `g` by computing the GCD in `\mathbb{Z}_p[x]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. Trial division is only made for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement univariate integer polynomial g : PolyElement univariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_univariate >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = x**5 - 1 >>> g = x - 1 >>> h, cff, cfg = modgcd_univariate(f, g) >>> h, cff, cfg (x - 1, x**4 + x**3 + x**2 + x + 1, 1) >>> cff * h == f True >>> cfg * h == g True >>> f = 6*x**2 - 6 >>> g = 2*x**2 + 4*x + 2 >>> h, cff, cfg = modgcd_univariate(f, g) >>> h, cff, cfg (2*x + 2, 3*x - 3, x + 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ Nrr&)rris_ZZr primitiver'r+rrrrr%rr6 quo_groundcontentdiv)rrresultrcfcgchboundr(mrrrr)r*hlastmhmhfquofremgquogremcffcfgrrrmodgcd_univariates^C    "          rKc Cs|j}|j}|j}i}|D]@\}}|dd|vrFi||dd<|||dd|d<qg}t|D]}t|t|||||}qp|j|j |dd} | | |} | | | |fS)a Compute the content and the primitive part of a polynomial in `\mathbb{Z}_p[x_0, \ldots, x_{k-2}, y] \cong \mathbb{Z}_p[y][x_0, \ldots, x_{k-2}]`. Parameters ========== f : PolyElement integer polynomial in `\mathbb{Z}_p[x0, \ldots, x{k-2}, y]` p : Integer modulus of `f` Returns ======= contf : PolyElement integer polynomial in `\mathbb{Z}_p[y]`, content of `f` ppf : PolyElement primitive part of `f`, i.e. `\frac{f}{contf}` Examples ======== >>> from sympy.polys.modulargcd import _primitive >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> p = 3 >>> f = x**2*y**2 + x**2*y - y**2 - y >>> _primitive(f, p) (y**2 + y, x**2 - 1) >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x*y*z - y**2*z**2 >>> _primitive(f, p) (z, x*y - y**2*z) Nr&symbols)rrngens itertermsitervaluesrrclonerN from_denserquoset_ring) rrrr kZcoeffsmonomr/contyringcontfrrr _primitives)r\cCsB|jj}d|d}|D] }|dd|kr|dd}q|S)a Compute the degree of a multivariate polynomial `f \in K[x_0, \ldots, x_{k-2}, y] \cong K[y][x_0, \ldots, x_{k-2}]`. Parameters ========== f : PolyElement polynomial in `K[x_0, \ldots, x_{k-2}, y]` Returns ======= degf : Integer tuple degree of `f` in `x_0, \ldots, x_{k-2}` Examples ======== >>> from sympy.polys.modulargcd import _deg >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _deg(f) (2,) >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _deg(f) (2, 2) >>> f = x*y*z - y**2*z**2 >>> _deg(f) (1, 1) )rr&NrL)rrOZ itermonoms)rrWdegfrXrrr_degZs (  r^c Cst|j}|j}|j|j|dd}|jd}t|}|j}|D],\}}|dd|krB||||d7}qB|S)a Compute the leading coefficient of a multivariate polynomial `f \in K[x_0, \ldots, x_{k-2}, y] \cong K[y][x_0, \ldots, x_{k-2}]`. Parameters ========== f : PolyElement polynomial in `K[x_0, \ldots, x_{k-2}, y]` Returns ======= lcf : PolyElement polynomial in `K[y]`, leading coefficient of `f` Examples ======== >>> from sympy.polys.modulargcd import _LC >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _LC(f) y**2 + y >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x**2*y**2 + x**2*y - 1 >>> _LC(f) 1 >>> f = x*y*z - y**2*z**2 >>> _LC(f) z r&rMrNrL)rrOrSrNr-r^rrP) rrrWrZyr]lcfrXr/rrr_LCs( racCsP|j}|j}|D]6\}}||f|d|||dd}|||<q|S)zS Make the variable `x_i` the leading one in a multivariate polynomial `f`. Nr&)rrrP)rr5rfswaprXr/Z monomswaprrr_swaps & rccCs0|j}|j|j|j}|jt|djt|dj}||}d}t|}||dkr`t|}qJ||}||}t||\} }t||\} }t| | |} | } tt |t ||} t |D]X}| d||sq| d||}| d||}t|||}| }|| fSt | | | fS)a Compute upper degree bounds for the GCD of two bivariate integer polynomials `f` and `g`. The GCD is viewed as a polynomial in `\mathbb{Z}[y][x]` and the function returns an upper bound for its degree and one for the degree of its content. This is done by choosing a suitable prime `p` and computing the GCD of the contents of `f \; \mathrm{mod} \, p` and `g \; \mathrm{mod} \, p`. The choice of `p` guarantees that the degree of the content in `\mathbb{Z}_p[y]` is greater than or equal to the degree in `\mathbb{Z}[y]`. To obtain the degree bound in the variable `x`, the polynomials are evaluated at `y = a` for a suitable `a \in \mathbb{Z}_p` and then their GCD in `\mathbb{Z}_p[x]` is computed. If no such `a` exists, i.e. the degree in `\mathbb{Z}_p[x]` is always smaller than the one in `\mathbb{Z}[y][x]`, then the bound is set to the minimum of the degrees of `f` and `g` in `x`. Parameters ========== f : PolyElement bivariate integer polynomial g : PolyElement bivariate integer polynomial Returns ======= xbound : Integer upper bound for the degree of the GCD of the polynomials `f` and `g` in the variable `x` ycontbound : Integer upper bound for the degree of the content of the GCD of the polynomials `f` and `g` in the variable `y` References ========== 1. [Monagan00]_ r&r)rrr'rrcrrr\r%rrar.evaluatemin)rrrgamma1gamma2 badprimesrrrcontfpcontgpconthp ycontbounddeltaafpagpahpaxboundrrr_degree_bound_bivariates0*        rsc Cst|}t|}||}|||||jjj}|jj}t|jjtr\t } ndd} |D]} | || || |||| <qh|D]} | || ||||| <q|D]} | ||| |||| <q|S)a: Construct a polynomial `h_{pq}` in `\mathbb{Z}_{p q}[x_0, \ldots, x_{k-1}]` such that .. math :: h_{pq} = h_p \; \mathrm{mod} \, p h_{pq} = h_q \; \mathrm{mod} \, q for relatively prime integers `p` and `q` and polynomials `h_p` and `h_q` in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` and `\mathbb{Z}_q[x_0, \ldots, x_{k-1}]` respectively. The coefficients of the polynomial `h_{pq}` are computed with the Chinese Remainder Theorem. The symmetric representation in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`, `\mathbb{Z}_q[x_0, \ldots, x_{k-1}]` and `\mathbb{Z}_{p q}[x_0, \ldots, x_{k-1}]` is used. Parameters ========== hp : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_p` hq : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_q` p : Integer modulus of `h_p`, relatively prime to `q` q : Integer modulus of `h_q`, relatively prime to `p` Examples ======== >>> from sympy.polys.modulargcd import _chinese_remainder_reconstruction_multivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> p = 3 >>> q = 5 >>> hp = x**3*y - x**2 - 1 >>> hq = -x**3*y - 2*x*y**2 + 2 >>> hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) >>> hpq 4*x**3*y + 5*x**2 + 3*x*y**2 + 2 >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True >>> R, x, y, z = ring("x, y, z", ZZ) >>> p = 6 >>> q = 5 >>> hp = 3*x**4 - y**3*z + z >>> hq = -2*x**4 + z >>> hpq = _chinese_remainder_reconstruction_multivariate(hp, hq, p, q) >>> hpq 3*x**4 + 5*y**3*z + z >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True cSst||g||gdddS)NTr,rr)cpZcqrr1rrrcrt_ksz<_chinese_remainder_reconstruction_multivariate..crt_) setmonoms intersectiondifference_updaterrr isinstancer._chinese_remainder_reconstruction_multivariate) r)r0rr1ZhpmonomsZhqmonomsrwrr4rurXrrrr{s"H      r{FcCs|j}|r |jj}|jj|}n|j}|j|}t||D]h\} } |j} |j} |D]&} | | kr`qR| || 9} | | | 9} qR|| |} | | }|| ||7}q:||S)a Reconstruct a polynomial `h_p` in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` from a list of evaluation points in `\mathbb{Z}_p` and a list of polynomials in `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, which are the images of `h_p` evaluated in the variable `x_i`. It is also possible to reconstruct a parameter of the ground domain, i.e. if `h_p` is a polynomial over `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`. In this case, one has to set ``ground=True``. Parameters ========== evalpoints : list of Integer objects list of evaluation points in `\mathbb{Z}_p` hpeval : list of PolyElement objects list of polynomials in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, images of `h_p` evaluated in the variable `x_i` ring : PolyRing `h_p` will be an element of this ring i : Integer index of the variable which has to be reconstructed p : Integer prime number, modulus of `h_p` ground : Boolean indicates whether `x_i` is in the ground domain, default is ``False`` Returns ======= hp : PolyElement interpolated polynomial in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` ) rrr-ziprrrrVr) evalpointshpevalrr5rgroundr)rr_rnrqZnumerdenombr/rrr_interpolate_multivariatexs$'    rc4CsF|j|jkr|jjjsJt||}|dur0|S|j}|\}}|\}}|j||}t||\}}||kr~dkrnn"||||||||fSt|d} t|d} | } | } t| | \} }| |krdkrnn"||||||||fS|j|j |j }|j| j | j }||}d}d}t |}||dkrht |}qN| |}| |}t ||\}}t ||\}}t|||}| }||krqFn||krd}|}qFtt|t||}| }| }| }t| || || ||d}||kr,qFd}g} g}!d}"t|D]}#|d|#}$|$|sbqD|d|# |}%|d|# |}&t|%|&|}'|' }(|(|krqDn|(|krd}|(}d}"q|'|$ |}'| |#|!|'|d7}||krDqqD|"rqF||krqFt| |!|d|})t |)|d})|)||})|) d}*|*| krbqF|*| krxd}|*} qF|)| |})|dkr|}|)}+qFt|)|+||},||9}|,|+ks|,}+qF|,|,}-||-\}.}/||-\}0}1|/sF|1sF|-j dkr| }|-|}-|.||}2|0||}3|-|2|3fSqFdS)a! Computes the GCD of two polynomials in `\mathbb{Z}[x, y]` using a modular algorithm. The algorithm computes the GCD of two bivariate integer polynomials `f` and `g` by calculating the GCD in `\mathbb{Z}_p[x, y]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. To compute the bivariate GCD over `\mathbb{Z}_p`, the polynomials `f \; \mathrm{mod} \, p` and `g \; \mathrm{mod} \, p` are evaluated at `y = a` for certain `a \in \mathbb{Z}_p` and then their univariate GCD in `\mathbb{Z}_p[x]` is computed. Interpolating those yields the bivariate GCD in `\mathbb{Z}_p[x, y]`. To verify the result in `\mathbb{Z}[x, y]`, trial division is done, but only for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement bivariate integer polynomial g : PolyElement bivariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_bivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 - y**2 >>> g = x**2 + 2*x*y + y**2 >>> h, cff, cfg = modgcd_bivariate(f, g) >>> h, cff, cfg (x + y, x - y, x + y) >>> cff * h == f True >>> cfg * h == g True >>> f = x**2*y - x**2 - 4*y + 4 >>> g = x + 2 >>> h, cff, cfg = modgcd_bivariate(f, g) >>> h, cff, cfg (x + 2, x*y - x - 2*y + 2, 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ Nrr&FT)rrr7rr8r'rsrrcrrrrr\r%rarer.rdappendrrVr{r9r:r;)4rrr<rr=r>r?rrrlrbZgswapdegyfdegygZyboundZ xcontboundrfrgrhrArrrrirjrkZ degconthprmZ degcontfpZ degcontgpdegdeltaNr2r}r~ZunluckyrndeltaarorprqZdeghpar)ZdegyhprBrCrDrErFrGrHrIrJrrrmodgcd_bivariatesI   "  "                           rc#Cs|j}|j}|dkrZt||||}|}||dkr>dS||dkrV||d<t|S||d} ||d} t||\} }t||\} }t| | |} | }| }| }|||dkrdS|||dkr|||d<tt|}t|}t|||}|}t|dD]*}|ttt ||tt |||9}q|}t | || |||d||d|d}||krdSd}d}g}g}t t|}|rt |dd}|||d||sڐq|d||}||d||}||d||} t|| |||}!|!durJ|d7}||krdSq|!jrf| ||}|S|!||}!||||!|d7}||krt||||d|}t||d| |}||d}"|"||dkrdS|"||dkr|"||d<t|SqdS)a Compute the GCD of two polynomials in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`. The algorithm reduces the problem step by step by evaluating the polynomials `f` and `g` at `x_{k-1} = a` for suitable `a \in \mathbb{Z}_p` and then calls itself recursively to compute the GCD in `\mathbb{Z}_p[x_0, \ldots, x_{k-2}]`. If these recursive calls are successful for enough evaluation points, the GCD in `k` variables is interpolated, otherwise the algorithm returns ``None``. Every time a GCD or a content is computed, their degrees are compared with the bounds. If a degree greater then the bound is encountered, then the current call returns ``None`` and a new evaluation point has to be chosen. If at some point the degree is smaller, the correspondent bound is updated and the algorithm fails. Parameters ========== f : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_p` g : PolyElement multivariate integer polynomial with coefficients in `\mathbb{Z}_p` p : Integer prime number, modulus of `f` and `g` degbound : list of Integer objects ``degbound[i]`` is an upper bound for the degree of the GCD of `f` and `g` in the variable `x_i` contbound : list of Integer objects ``contbound[i]`` is an upper bound for the degree of the content of the GCD in `\mathbb{Z}_p[x_i][x_0, \ldots, x_{i-1}]`, ``contbound[0]`` is not used can therefore be chosen arbitrarily. Returns ======= h : PolyElement GCD of the polynomials `f` and `g` or ``None`` References ========== 1. [Monagan00]_ 2. [Brown71]_ r&rN)rrOr%rrr r\rar.rcrelistrandomsampleremoverd_modgcd_multivariate_pZ is_groundrVrrr)#rrrdegbound contboundrrWrDdeghrrr[ZcontgZconthZdegcontfZdegcontgZdegconthr`ZlcgrmZevaltestr5rrr2dr}hevalpointsrnrfagahaZdegyhrrrrs0     (         rcCs |j|jkr|jjjsJt||}|dur0|S|j}|j}|\}}|\}}|j||}|j|j|j}|jj} t |D]&} | |jt || jt || j9} qddt | | D} t | } d} d}t|}| |dkrt|}q||}||}zt|||| | }Wnty:d} YqYn0|durHq|||}| dkrl|} |}qt|||| }| |9} ||ks|}q|d}||\}}||\}}|s|s|jdkr| }||}|||}|||}|||fSqdS)a Compute the GCD of two polynomials in `\mathbb{Z}[x_0, \ldots, x_{k-1}]` using a modular algorithm. The algorithm computes the GCD of two multivariate integer polynomials `f` and `g` by calculating the GCD in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem. To compute the multivariate GCD over `\mathbb{Z}_p` the recursive subroutine :func:`_modgcd_multivariate_p` is used. To verify the result in `\mathbb{Z}[x_0, \ldots, x_{k-1}]`, trial division is done, but only for candidates which are very likely the desired GCD. Parameters ========== f : PolyElement multivariate integer polynomial g : PolyElement multivariate integer polynomial Returns ======= h : PolyElement GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac{f}{h}` cfg : PolyElement cofactor of `g`, i.e. `\frac{g}{h}` Examples ======== >>> from sympy.polys.modulargcd import modgcd_multivariate >>> from sympy.polys import ring, ZZ >>> R, x, y = ring("x, y", ZZ) >>> f = x**2 - y**2 >>> g = x**2 + 2*x*y + y**2 >>> h, cff, cfg = modgcd_multivariate(f, g) >>> h, cff, cfg (x + y, x - y, x + y) >>> cff * h == f True >>> cfg * h == g True >>> R, x, y, z = ring("x, y, z", ZZ) >>> f = x*z**2 - y*z**2 >>> g = x**2*z + z >>> h, cff, cfg = modgcd_multivariate(f, g) >>> h, cff, cfg (z, x*z - y*z, x**2 + 1) >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Monagan00]_ 2. [Brown71]_ See also ======== _modgcd_multivariate_p NcSsg|]\}}t||qSr)re).0ZfdegZgdegrrr z'modgcd_multivariate..r&r)rrr7rrOr8r'rrr.rcr|degreesrrrrr rr{r;)rrr<rrWr=r>r?r(rhr5rrrArrrr)rBrCrDrErFrGrHrIrJrrrmodgcd_multivariate&sbN    $           rcCs6|j}t||||j\}}||||fS)z_ Compute `\frac f g` modulo `p` for two univariate polynomials over `\mathbb Z_p`. )rr to_denserrT)rrrrZdensequoZdenseremrrr_gf_divsrcCs |j}|j}|}|d}||d}||j}} ||j} } | |krt|| |d} | || | |}} | | | | |} } q@| | } }||kst|||dkrdS|j}|dkr| ||}| ||} | ||}| }|| ||S)a  Reconstruct a rational function `\frac a b` in `\mathbb Z_p(t)` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are polynomials in `\mathbb Z_p[t]` and `m` has positive degree. The algorithm is based on the Euclidean Algorithm. In general, `m` is not irreducible, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : PolyElement univariate polynomial in `\mathbb Z[t]` p : Integer prime number m : PolyElement modulus, not necessarily irreducible Returns ======= frac : FracElement either `\frac a b` in `\mathbb Z(t)` or ``None`` References ========== 1. [Hoeij04]_ r&rN) rrrrrrrr%rrrto_field)crrArrMrDr0s0r1s1rUrnrlcr#fieldrrr!_rational_function_reconstructions*%      rc Cs|j}|D]p\}}|dkr6t|||}|svdSn@|jj}||D](\} } t| ||} | sldS| || <qL|||<q|S)a Reconstruct every coefficient `c_h` of a polynomial `h` in `\mathbb Z_p(t_k)[t_1, \ldots, t_{k-1}][x, z]` from the corresponding coefficient `c_{h_m}` of a polynomial `h_m` in `\mathbb Z_p[t_1, \ldots, t_k][x, z] \cong \mathbb Z_p[t_k][t_1, \ldots, t_{k-1}][x, z]` such that .. math:: c_{h_m} = c_h \; \mathrm{mod} \, m, where `m \in \mathbb Z_p[t]`. The reconstruction is based on the Euclidean Algorithm. In general, `m` is not irreducible, so it is possible that this fails for some coefficient. In that case ``None`` is returned. Parameters ========== hm : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][x, z]` p : Integer prime number, modulus of `\mathbb Z_p` m : PolyElement modulus, polynomial in `\mathbb Z[t]`, not necessarily irreducible ring : PolyRing `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]`, `h` will be an element of this ring k : Integer index of the parameter `t_k` which will be reconstructed Returns ======= h : PolyElement reconstructed polynomial in `\mathbb Z(t_k)[t_1, \ldots, t_{k-1}][x, z]` or ``None`` See also ======== _rational_function_reconstruction rN)rrPrrdrop_to_ground) rCrrArrWrDrXr/coeffhmonrr?rrr$_rational_reconstruction_func_coeffss.    rcCs@|j}t||||j\}}}||||||fS)z Extended Euclidean Algorithm for two univariate polynomials over `\mathbb Z_p`. Returns polynomials `s, t` and `h`, such that `h` is the GCD of `f` and `g` and `sf + tg = h \; \mathrm{mod} \, p`. )rr rrrT)rrrrstrDrrr _gf_gcdexKs rcCs4|j}||}||}||||g|S)a Compute the reduced representation of a polynomial `f` in `\mathbb Z_p[z] / (\check m_{\alpha}(z))[x]` Parameters ========== f : PolyElement polynomial in `\mathbb Z[x, z]` minpoly : PolyElement polynomial `\check m_{\alpha} \in \mathbb Z[z]`, not necessarily irreducible p : Integer prime number, modulus of `\mathbb Z_p` Returns ======= ftrunc : PolyElement polynomial in `\mathbb Z[x, z]`, reduced modulo `\check m_{\alpha}(z)` and `p` )rrVZ ground_newrr!)rminpolyrrZp_rrr_truncYs  rc Cs|j}t|||}t|||}|r|}|d}t||||\}}} | dksTdS|d} | |krhq||||} t||| |df| ||}qT|}|}qt||||d|} t|| ||S)a  Compute the monic GCD of two univariate polynomials in `\mathbb{Z}_p[z]/(\check m_{\alpha}(z))[x]` with the Euclidean Algorithm. In general, `\check m_{\alpha}(z)` is not irreducible, so it is possible that some leading coefficient is not invertible modulo `\check m_{\alpha}(z)`. In that case ``None`` is returned. Parameters ========== f, g : PolyElement polynomials in `\mathbb Z[x, z]` minpoly : PolyElement polynomial in `\mathbb Z[z]`, not necessarily irreducible p : Integer prime number, modulus of `\mathbb Z_p` Returns ======= h : PolyElement GCD of `f` and `g` in `\mathbb Z[z, x]` or ``None``, coefficients are in `\left[ -\frac{p-1} 2, \frac{p-1} 2 \right]` rr&N)rrrrdmp_LCrVr) rrrrrr!r"r#_r'r$rUZlcfinvrrr_euclidean_algorithmxs$    $rc Cs |j}|j|jd|jdfd}||}|}|}|}|d} t||} |j} |r||krt||} || |||df| }|r||}|d}|r|| krt|||} | | |d|| f| }|r||}|d}q|}q^|S)a= Check if `h` divides `f` in `\mathbb K[t_1, \ldots, t_k][z]/(m_{\alpha}(z))`, where `\mathbb K` is either `\mathbb Q` or `\mathbb Z_p`. This algorithm is based on pseudo division and does not use any fractions. By default `\mathbb K` is `\mathbb Q`, if a prime number `p` is given, `\mathbb Z_p` is chosen instead. Parameters ========== f, h : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement polynomial `m_{\alpha}(z)` in `\mathbb Z[t_1, \ldots, t_k][z]` p : Integer or None if `p` is given, `\mathbb K` is set to `\mathbb Z_p` instead of `\mathbb Q`, default is ``None`` Returns ======= rem : PolyElement remainder of `\frac f h` References ========== .. [1] [Hoeij02]_ r&rrM) rrSrNrVrrarrrr) rrDrrrZzxringr!r$rZdegmZlchlcmZlcremrrr_trial_divisions.!        rcCsF|jj|jjj|d}|j}|D]\}}|||||<q(|S)z[ Evaluate a polynomial `f` at `a` in the `i`-th variable of the ground domain. r)rrSrdroprrPrd)rr5rnrrrXr/rrr_evaluate_grounds rc& Cs|j}|j}t|tr|j}nt||||S|dkr@|j}n$|j|d}|j|jjd}|j|d}d} d} | || |} |j } g} g}g}t t |}|rt |dd}|||dkr| |d||dk}n| |d||dk}|rqt| |d|}t||d|}||||gdkrJqt||d|}t||d|}t||||}|dur| d7} | | krdSq|dkr|S|gdg|d}|dkr|D]B\}}|d|dkr|jt|ddkr|j|dd<q|g}|g}|dkr8|jj}n|jjj}|jjd}t| ||D]6\}} }!|!|kr^|||| |||9}q^||}| |||||| d7} t||||d|dd}"t|"||||d}"|"durq|dkrL|jj }#|#jj}$|"!D](}|#j"t#|$$|j%$||#j}$q nT|jjj }#|#jj}$|"!D]8}|!D](}%|#j"t#|$$|%j%$||#j}$qrqf|&|$|}$||"'|$(|}"t)||"||st)||"||s|"SqdS)a Compute the GCD of two polynomials `f` and `g` in `\mathbb Z_p(t_1, \ldots, t_k)[z]/(\check m_\alpha(z))[x]`. The algorithm reduces the problem step by step by evaluating the polynomials `f` and `g` at `t_k = a` for suitable `a \in \mathbb Z_p` and then calls itself recursively to compute the GCD in `\mathbb Z_p(t_1, \ldots, t_{k-1})[z]/(\check m_\alpha(z))[x]`. If these recursive calls are successful, the GCD over `k` variables is interpolated, otherwise the algorithm returns ``None``. After interpolation, Rational Function Reconstruction is used to obtain the correct coefficients. If this fails, a new evaluation point has to be chosen, otherwise the desired polynomial is obtained by clearing denominators. The result is verified with a fraction free trial division. Parameters ========== f, g : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][z]`, not necessarily irreducible p : Integer prime number, modulus of `\mathbb Z_p` Returns ======= h : PolyElement primitive associate in `\mathbb Z[t_1, \ldots, t_k][x, z]` of the GCD of the polynomials `f` and `g` or ``None``, coefficients are in `\left[ -\frac{p-1} 2, \frac{p-1} 2 \right]` References ========== 1. [Hoeij04]_ r&rrNT)r)*rrrzrrOrrrrSrrrr.rrrrdrrr!_func_field_modgcd_prrPLMtupleget_ringrr-r|rrrr itercoeffsrTr rrZ domain_newras_exprr)&rrrrrrrWZqdomainZqringr2rr(rmr}rLMlistrrntestZgammaaZminpolyarrrrrXr/Z evalpoints_aZheval_arArrZhbZLMhbrDr denrrrrrs*        *                rc Cs|dkr||7}||j}}||j}}t|d}||krj||}||||}}||||}}q4t||krzdS|dkr| | } } n|dkr||} } ndS|} | | | | S)a Reconstruct a rational number `\frac a b` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are integers. The algorithm is based on the Euclidean Algorithm. In general, `m` is not a prime number, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : Integer `c = \frac a b \; \mathrm{mod} \, m` m : Integer modulus, not necessarily prime domain : IntegerRing `a, b, c` are elements of ``domain`` Returns ======= frac : Rational either `\frac a b` in `\mathbb Q` or ``None`` References ========== 1. [Wang81]_ rrN)rrr abs get_field) rrArrrrrr@rUrnrrrrr _integer_rational_reconstructions$$     rc Cs`|j}t|jtr t}|jj}n t}|jj}|D]&\}}||||}|sRdS|||<q4|S)a Reconstruct every rational coefficient `c_h` of a polynomial `h` in `\mathbb Q[t_1, \ldots, t_k][x, z]` from the corresponding integer coefficient `c_{h_m}` of a polynomial `h_m` in `\mathbb Z[t_1, \ldots, t_k][x, z]` such that .. math:: c_{h_m} = c_h \; \mathrm{mod} \, m, where `m \in \mathbb Z`. The reconstruction is based on the Euclidean Algorithm. In general, `m` is not a prime number, so it is possible that this fails for some coefficient. In that case ``None`` is returned. Parameters ========== hm : PolyElement polynomial in `\mathbb Z[t_1, \ldots, t_k][x, z]` m : Integer modulus, not necessarily prime ring : PolyRing `\mathbb Q[t_1, \ldots, t_k][x, z]`, `h` will be an element of this ring Returns ======= h : PolyElement reconstructed polynomial in `\mathbb Q[t_1, \ldots, t_k][x, z]` or ``None`` See also ======== _integer_rational_reconstruction N)rrzrr#_rational_reconstruction_int_coeffsrrrP) rCrArrDZreconstructionrrXr/rrrrrs)    rcCs|j}|j}t|tr>|j}|jj|jd}|j|d}nd}|j|jd}|\}}|\} }||||} |j } d} g} g}g}t | } | | dkrq|dkr| | dk}n| | dk}|rq| | }| | }| | }t |||| }|durq|dkr |j S|gdg|}|dkr|D]B\}}|d|dkrF|jt|ddkrF|j|dd<qF|}| }t| ||D],\}}}||krt||||}||9}q| | ||||t|||}|durq|dkr|d}n8|jj }|D]}|j||d}q*||}||}|d}t||||st|| ||s|SqdS)a Compute the GCD of two polynomials in `\mathbb Q(t_1, \ldots, t_k)[z]/(m_{\alpha}(z))[x]` using a modular algorithm. The algorithm computes the GCD of two polynomials `f` and `g` by calculating the GCD in `\mathbb Z_p(t_1, \ldots, t_k)[z] / (\check m_{\alpha}(z))[x]` for suitable primes `p` and the primitive associate `\check m_{\alpha}(z)` of `m_{\alpha}(z)`. Then the coefficients are reconstructed with the Chinese Remainder Theorem and Rational Reconstruction. To compute the GCD over `\mathbb Z_p(t_1, \ldots, t_k)[z] / (\check m_{\alpha})[x]`, the recursive subroutine ``_func_field_modgcd_p`` is used. To verify the result in `\mathbb Q(t_1, \ldots, t_k)[z] / (m_{\alpha}(z))[x]`, a fraction free trial division is used. Parameters ========== f, g : PolyElement polynomials in `\mathbb Z[t_1, \ldots, t_k][x, z]` minpoly : PolyElement irreducible polynomial in `\mathbb Z[t_1, \ldots, t_k][z]` Returns ======= h : PolyElement the primitive associate in `\mathbb Z[t_1, \ldots, t_k][x, z]` of the GCD of `f` and `g` Examples ======== >>> from sympy.polys.modulargcd import _func_field_modgcd_m >>> from sympy.polys import ring, ZZ >>> R, x, z = ring('x, z', ZZ) >>> minpoly = (z**2 - 2).drop(0) >>> f = x**2 + 2*x*z + 2 >>> g = x + z >>> _func_field_modgcd_m(f, g, minpoly) x + z >>> D, t = ring('t', ZZ) >>> R, x, z = ring('x, z', D) >>> minpoly = (z**2-3).drop(0) >>> f = x**2 + (t + 1)*x*z + 3*t >>> g = x*z + 3*t >>> _func_field_modgcd_m(f, g, minpoly) x + t*z References ========== 1. [Hoeij04]_ See also ======== _func_field_modgcd_p rrr&N)rrrzrrOrSrr8rrrrrrrrPrrr|r{rrZ clear_denomsrrrrVr)rrrrrrWZQQdomainZQQringr=r>r(rmrZprimesZhplistrrrrZminpolypr)rrXr/rCrAr1r0ZLMhqrDrrrr_func_field_modgcd_m&szB         *            rc Cs|j}t|jtr|jj}n|j}|j}|D]"}|jD]}|r:|||j}q:q0| D]\}}|j}|jj}t|jtr| |dd}t |} t | D]r} || r||| ||}|d| | df|vr|||d| | df<q||d| | df|7<qq\|S)a Compute an associate of a polynomial `f \in \mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` in `\mathbb Z[x_1, \ldots, x_{n-1}][z] / (\check m_{\alpha}(z))[x_0]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` ring : PolyRing `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` Returns ======= f_ : PolyElement associate of `f` in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` r&Nr) rrzrrrrrepr denominatorrPrlenr.) rrf_rrr/rrXrAr2r5rrr _to_ZZ_polys,      $rc Cs|j}|j}t|jjtr|D]h\}}|D]V\}}|df|}|||gdg|d} ||vrx| ||<q2||| 7<q2q"n\|D]R\}}|df}|||gdg|d} ||vr| ||<q||| 7<q|S)ar Convert a polynomial `f \in \mathbb Z[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha}(z))[x_0]` to a polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` ring : PolyRing `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= f_ : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` rr&)rrrzrrrP) rrrrrXr/rZcoefrArrrr _to_ANP_polys"   rcCs*|j}|D]\}}||||<q|S)zo Change representation of the minimal polynomial from ``DMP`` to ``PolyElement`` for a given ring. )rZtermsr)rrZminpoly_rXr/rrr_minpoly_from_dense/srcCsx|j}|jtd|j}|jj}||}|j}|D](}t||d}||j kr:||fSq:|| | |fS)z Compute the content in `x_0` and the primitive part of a polynomial `f` in `\mathbb Q(\alpha)[x_0, x_1, \ldots, x_{n-1}] \cong \mathbb Q(\alpha)[x_1, \ldots, x_{n-1}][x_0]`. r&r) rrr.rOrrrrfunc_field_modgcdrrUrV)rZfringrr rrYr/rrr_primitive_in_x0<s   rcCsr|j}|j}|j}||jkr"|js&Jt||}|dur<|Std}|j|j|f|jd}|dkrt ||}t ||} | d |j j } t|| | } t| |} nt|\} }t|\} }t| | d}|jtd|}t ||}t ||} t|j | d} t|| | } t| |} t| \}} | ||9} || |9}|| |9}| | j} | || || fS)a Compute the GCD of two polynomials `f` and `g` in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` using a modular algorithm. The algorithm first computes the primitive associate `\check m_{\alpha}(z)` of the minimal polynomial `m_{\alpha}` in `\mathbb{Z}[z]` and the primitive associates of `f` and `g` in `\mathbb{Z}[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha})[x_0]`. Then it computes the GCD in `\mathbb Q(x_1, \ldots, x_{n-1})[z]/(m_{\alpha}(z))[x_0]`. This is done by calculating the GCD in `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]` for suitable primes `p` and then reconstructing the coefficients with the Chinese Remainder Theorem and Rational Reconstuction. The GCD over `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]` is computed with a recursive subroutine, which evaluates the polynomials at `x_{n-1} = a` for suitable evaluation points `a \in \mathbb Z_p` and then calls itself recursively until the ground domain does no longer contain any parameters. For `\mathbb{Z}_p[z]/(\check m_{\alpha}(z))[x_0]` the Euclidean Algorithm is used. The results of those recursive calls are then interpolated and Rational Function Reconstruction is used to obtain the correct coefficients. The results, both in `\mathbb Q(x_1, \ldots, x_{n-1})[z]/(m_{\alpha}(z))[x_0]` and `\mathbb{Z}_p(x_1, \ldots, x_{n-1})[z]/(\check m_{\alpha}(z))[x_0]`, are verified by a fraction free trial division. Apart from the above GCD computation some GCDs in `\mathbb Q(\alpha)[x_1, \ldots, x_{n-1}]` have to be calculated, because treating the polynomials as univariate ones can result in a spurious content of the GCD. For this ``func_field_modgcd`` is called recursively. Parameters ========== f, g : PolyElement polynomials in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= h : PolyElement monic GCD of the polynomials `f` and `g` cff : PolyElement cofactor of `f`, i.e. `\frac f h` cfg : PolyElement cofactor of `g`, i.e. `\frac g h` Examples ======== >>> from sympy.polys.modulargcd import func_field_modgcd >>> from sympy.polys import AlgebraicField, QQ, ring >>> from sympy import sqrt >>> A = AlgebraicField(QQ, sqrt(2)) >>> R, x = ring('x', A) >>> f = x**2 - 2 >>> g = x + sqrt(2) >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == x + sqrt(2) True >>> cff * h == f True >>> cfg * h == g True >>> R, x, y = ring('x, y', A) >>> f = x**2 + 2*sqrt(2)*x*y + 2*y**2 >>> g = x + sqrt(2)*y >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == x + sqrt(2)*y True >>> cff * h == f True >>> cfg * h == g True >>> f = x + sqrt(2)*y >>> g = x + y >>> h, cff, cfg = func_field_modgcd(f, g) >>> h == R.one True >>> cff * h == f True >>> cfg * h == g True References ========== 1. [Hoeij04]_ Nz)rNrr&r)rrrOZ is_AlgebraicrrrSrNrrrrTmodrrrrrrr.rrVr9rrU)rrrrr2r<rZZZringrZg_rrDZcontx0fZcontx0gZcontx0hZZZring_Zcontx0h_rrrrQs<h             r)F)N)3Zsympy.core.symbolrZ sympy.ntheoryrZsympy.ntheory.modularrZsympy.polys.domainsrZsympy.polys.galoistoolsrrr r r Zsympy.polys.polyerrorsr Zmpmathr rrr%r+r6rKr\r^rarcrsr{rrrrrrrrrrrrrrrrrrrrrrrrrsZ      !A=05 Kb AU BF8 E'@=:3