\hi8SSKJr SSKJr SSKJr SSKJr SSKJ r J r J r J r J r SSKJr SSKJr SSKrS rS rS rS rS rSrSrSrSrSrSrS)SjrSrSr Sr!Sr"Sr#Sr$Sr%Sr&Sr'S*Sjr(Sr)S r*S!r+S"r,S#r-S$r.S%r/S&r0S'r1S(r2g)+)Dummy) nextprimecrt)PolynomialRing)gf_gcd gf_from_dictgf_gcdexgf_divgf_lcm)ModularGCDFailed)sqrtNcURnU(d*U(d#URURUR4$U(dWURURR:aU*URUR*4$XRUR4$U(dWURURR:aU*UR*UR4$XRUR4$g)zb Compute the GCD of two polynomials in trivial cases, i.e. when one or both polynomials are zero. N)ringzeroLCdomainone)fgrs N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/modulargcd.py _trivial_gcdr s 66D yy$))TYY..  44$++"" "2tyy488)+ +ii) )  44$++"" "2y$))+ +hh ) ) cURRnU(aUnUR5nURURU5nUR5nXu:aOBXAR Xu- 45R XdR-5- RU5nMXUnUnU(aMUR URURU55RU5$)zE Compute the GCD of two univariate polynomials in `\mathbb{Z}_p[x]`. )rrdegreeinvertr mul_monom mul_ground trunc_ground)fpgppdomremdeglcinvdegrems r_gf_gcdr(#s ''..C iik 255!$ZZ\F|v|o6AA%&&.QQ__`abC    " ==BEE1- . ; ;A >>rcJURRRURUR5nSn[ U5nX#-S:Xa[ U5nX#-S:XaMUR U5nUR U5n[ XEU5nUR5nU$)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)rrgammar"r r!hpdeghps r_degree_bound_univariater/:s& FFMM  addADD )E A! A )q. aL )q.  B  B  B IIKE Lrc :UR5nURRSnURRn[ US-5H9n[ X#/UR XW-5UR XW-5/SS9SXg4'M; UR5 U$)a Construct a polynomial `h_{pq}` in `\mathbb{Z}_{p q}[x]` such that .. math :: h_{pq} = h_p \; \mathrm{mod} \, p h_{pq} = h_q \; \mathrm{mod} \, q for relatively prime integers `p` and `q` and polynomials `h_p` and `h_q` in `\mathbb{Z}_p[x]` and `\mathbb{Z}_q[x]` respectively. The coefficients of the polynomial `h_{pq}` are computed with the Chinese Remainder Theorem. The symmetric representation in `\mathbb{Z}_p[x]`, `\mathbb{Z}_q[x]` and `\mathbb{Z}_{p q}[x]` is used. It is assumed that `h_p` and `h_q` have the same degree. Parameters ========== hp : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_p` hq : PolyElement univariate integer polynomial with coefficients in `\mathbb{Z}_q` p : Integer modulus of `h_p`, relatively prime to `q` q : Integer modulus of `h_q`, relatively prime to `p` Examples ======== >>> from sympy.polys.modulargcd import _chinese_remainder_reconstruction_univariate >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> p = 3 >>> q = 5 >>> hp = -x**3 - 1 >>> hq = 2*x**3 - 2*x**2 + x >>> hpq = _chinese_remainder_reconstruction_univariate(hp, hq, p, q) >>> hpq 2*x**3 + 3*x**2 + 6*x + 5 >>> hpq.trunc_ground(p) == hp True >>> hpq.trunc_ground(q) == hq True rr*T symmetric)rrgensrrangercoeff strip_zero)r-hqr"qnxhpqis r,_chinese_remainder_reconstruction_univariater=[sl A  QA '',,C 1Q3Z!$!$ @DQRSTD NN JrcURUR:Xa%URRR(de[X5nUbU$URnUR 5up@UR 5upQURR XE5n[ X5nUS:Xa-U"U5URXF-5URXV-54$URR URUR5nSn Sn [U 5n X-S:Xa[U 5n X-S:XaMURU 5n URU 5n [XU 5n U R5nX:aMnX:aSn UnMyU RU5RU 5n U S:XaU n U nM[U WX5nX-n UU:XdUnMURUR55nUR!U5unnUR!U5unnU(dVU(dOURS:aU*nURU5nURXF-5nURXV-5nUUU4$GMf)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]_ rr*)rris_ZZr primitiver+r/rrrrr(rr= quo_groundcontentdiv)rrresultrcfcgchboundr,mr"r r!r-r.hlastmhmhfquofremgquogremcffcfgs rmodgcd_univariaterSs)F 66QVV  3 33 3 ! F  66D KKMEB KKMEB  B $Q *E zBxbh/bh1GGG KKOOADD!$$ 'E A A  aLi1n! Ai1n^^A  ^^A  RQ   =  ]AE  ]]5 ! . .q 1 6AF  9"fa K V|F  MM"**, 'UU1X dUU1X dDttaxS R A//"(+C//"(+Cc3; O rc URnURnURn0nUR5H"upgUSSU;a0XVSS'XuUSSUS'M$ /n[ UR 55Hn[ U[XqU5X5nM URURUS- S9n U RU5RU5n XRU RU554$)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 itertermsitervaluesrr clonerW from_denserquoset_ring) rr"rr#kcoeffsmonomr5contyringcontfs r _primitiverfsR 66D ++C A F  ":V #!#F": (-uSbz59%& Dfmmo&dL37@' JJt||AaC0J 1E   T " / / 2E %%t,- --rcURRnSUS- -nUR5HnUSSU:dMUSSnM U$)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*NrU)rrX itermonoms)rr`degfrbs r_degrjZsOP  A 1Q3>> 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*rVrNrU)rrXr\rWr3rjrrY) rrr`rdyrilcfrbr5s r_LCrnsP 66D A JJt||AaC0J 1E 1 A 7D **C  ":  "I% %C& JrcURnURnUR5HupEXA4USU-XAS-S-nXSU'M U$)zK Make the variable `x_i` the leading one in a multivariate polynomial `f`. Nr*)rrrY)rr<rfswaprbr5 monomswaps r_swaprrsX 66D IIE XK%)+eaCDk9  i& LrcURnURRURUR5nURR[ US5R[ US5R5nX4-nSn[ U5nXV-S:Xa[ U5nXV-S:XaMUR U5nUR U5n[Xv5up[X5up[XU5n U R5n [[U5[U5U5n [U5HnU RSU5U-(dMURSU5R U5nURSU5R U5n[UUU5nUR5nUU 4s $ [UR5UR55U 4$)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+rrrrrrfr(rrnr4evaluatemin)rrrgamma1gamma2 badprimesr"r r!contfpcontgpconthp ycontbounddeltaafpagpahpaxbounds r_degree_bound_bivariatersT 66D [[__QTT144 (F [[__U1a[^^U1a[^^ >> 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 c0>T"[X#/X/SS9S5$)NTr1rr)cpcqr"r8rs rcrt_<_chinese_remainder_reconstruction_multivariate..crt_ls #qfrh$?BC Cr) setmonoms intersectiondifference_updaterrr isinstancer._chinese_remainder_reconstruction_multivariate) r-r7r"r8hpmonomshqmonomsrrr;rrbrs @rrrsP299;H299;H  " "8 ,F v& v& WW^^F ;;D '',,C"''...11= D")RY5 ")Ta0 $5 10  JrcURnU(a0URRnURRUnOURnURUn[X5HsupURn URn UHn X:XaM XU - -n XU - -n M UR X5n U R U 5nXjRU5U-- nMu URU5$)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}]` ) rrr3ziprrrr_r) evalpointshpevalrr<r"groundr-rrlr~rnumerdenombr5s r_interpolate_multivariaterysN B ## KK  Q  IIaLj) Av UNE UNE  e'  ' ll4 5((* ??1 rcr URUR:Xa%URRR(de[X5nUbU$URnUR 5up@UR 5upQURR XE5n[ X5upxXxs=:XaS:Xa0O O-U"U5URXF-5URXV-54$[US5n [US5n U R5n U R5n [ X5upXs=:XaS:Xa0O O-U"U5URXF-5URXV-54$URR URUR5nURR U RU R5nUU-nSnSn[U5nUU-S:Xa[U5nUU-S:XaMURU5nURU5n[UU5unn[UU5unn[UUU5nUR5nUU:aMUU:aSnUnM[[U5[U5U5nUR5nUR5nUR5n[!U U- U U- X- U-5S-nUU:aGMSn/n /n!Sn"[#U5Hn#UR%SU#5n$U$U-(dM!UR%SU#5RU5n%UR%SU#5RU5n&[U%U&U5n'U'R5n(U(U:aMU(U:aSnU(nSn" OSU'RU$5RU5n'U R'U#5 U!R'U'5 US- nUU:XdM O U"(aGMUU:aGM[)U U!USU5n)[U)U5Sn)U)UR+U5-n)U)RS5n*U*U :aGMjU*U :aSnU*n GMwU)RU5RU5n)US:XaUnU)n+GM[-U)W+UU5n,UU-nU,U+:XdU,n+GMU,R/U,R155n-UR3U-5un.n/UR3U-5un0n1U/(dVU1(dOU-RS:aU*nU-RU5n-U.RXF-5n2U0RXV-5n3U-U2U34$GMh)aI 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]_ rr*TF)rrr?rr@r+rrrrrrrrrfr(rnrur4rtappendrr_rrArBrC)4rrrDrrErFrGrr|rpgswapdegyfdegygybound xcontboundrvrwrxrIr"r r!ryrzr{ degconthpr} degcontfp degcontgpdegdeltaNr9rrunluckyr~deltaarrrdeghpar-degyhprJrKrLrMrNrOrPrQrRs4 rmodgcd_bivariatersR 66QVV  3 33 3 ! F  66D KKMEB KKMEB  B06F  q Bxbh/bh1GGG !QKE !QKE LLNE LLNE0>F  q Bxbh/bh1GGG[[__QTT144 (F [[__UXXuxx 0FI A A  aL!mq ! A!mq ^^A  ^^A A& A& +MMO z !   #A"J BR!,MMO MMO <<>  !59#4  ( * ,./ 0 q5   qA^^Aq)FA:++a#003C++a#003C#sA&CZZ\F&..(55a8C   a MM#  FAAv14   q5  &z64A F A q ! &//$' '1 F?  F?AF  ]]6 " / / 2 6AF  ;B1 M QV|F  MM"**, 'UU1X dUU1X dDttaxS R A//"(+C//"(+Cc3; O rc "URnURnUS:XaH[XU5RU5nUR 5nXS:agXS:a XS'[ eU$UR US- 5n UR US- 5n [ X5up[ X5up[XU5n U R 5nU R 5nU R 5nUXFS- :agUXFS- :a UXFS- '[ e[U5n[U5n[UUU5nUn[US- 5H9nU[[[UU55[[UU55U5-nM; UR 5n[X- X- X6S- XFS- - U-5S-nUU:agSnSn/n/n[[U55nU(Ga[R"US5SnURU5 URSU5U-(dMOURSU5U-nURUS- U5RU5nURUS- U5RU5n [!UU X#U5n!U!cUS- nUU:agMU!R"(a"U R%U5RU5nU$U!R'U5RU5n!UR)U5 UR)U!5 US- nUU:Xak[+UUXVS- U5n[ Xr5SU R%U5-nUR US- 5n"U"X6S- :agU"X6S- :a U"X6S- '[ eU$U(aGMg)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)rrXr(rrr rfrnr4rrrulistrandomsampleremovert_modgcd_multivariate_p is_groundr_rrr)#rrr"degbound contboundrr`rLdeghrrrecontgconthdegcontfdegcontgdegconthrmlcgr}evaltestr<rrr9drhevalpointsr~rfagahadegyhs# rrrsx` 66D AAv A!  ) )! ,xxz 1+  1+ QK" " HHQqSME HHQqSME!HE!HE E! $E||~H||~H||~H)aC. )aC. ! A# a&C a&C Ca EH 1Q3ZGCa ,c%1+.>BB||~H E e. qSMIcN *X 5 79: ;A 1u A AJ E %(^F  MM&! $Q ' a  A&* 1%) ZZ!Q  , ,Q / ZZ!Q  , ,Q /$BA C : FA1u  <<t$11!4AH ]]6 " / / 2! R Q 6)*eTQ3JA1 #ennT&::AHHQqSMEx!}$x!}$ %1 &&HW &Z rc URUR:Xa%URRR(de[X5nUbU$URnURnUR 5upPUR 5upaURR XV5nURR URUR5nURRn [U5HGn XRR [X 5R[X5R5-n MI [UR5UR55V V s/sHup[X5PM n n n [U 5nSnSn[U5nU U-S:Xa[U5nU U-S:XaMUR!U5nUR!U5n[#UUUX5nUcMaUR'U5R!U5nUS:XaUnUnM[)UWUU5nUU-nUU:XdUnMUR 5SnUR+U5unnUR+U5unnU(dVU(dOURS:aU*nUR'U5nUR'XW-5nUR'Xg-5nUUU4$GMDs sn n f![$a SnGM[f=f)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 r*r)rrr?rrXr@r+rrr4rrrdegreesrurrrrr rrrC)rrrDrr`rErFrGr,rxr<fdeggdegrrrIr"r r!r-rJrKrLrMrNrOrPrQrRs rmodgcd_multivariater's\ 66QVV  3 33 3 ! F  66D A KKMEB KKMEB  B KKOOADD!$$ 'E I 1X[[__U1[^^U1[^^DD 36aiik199;2OP2OJDD2OHPXI A A  aL!mq ! A!mq ^^A  ^^A  'B8GB :  ]]5 ! . .q 1 6AF  ;B1 M QV|F  LLN1 UU1X dUU1X dDttaxS R A//"(+C//"(+Cc3; Q  Q"  A  s0K'K$$ K54K5cURn[UR5UR5X#R5upEUR U5UR U54$)zS Compute `\frac f g` modulo `p` for two univariate polynomials over `\mathbb Z_p`. )rr to_denserr])rrr"rdensequodenserems r_gf_divrsK 66D ajjlA{{KH ??8 $dooh&? ??rcURnURnUR5nUS-nXV- S- nX#RpXRpU R5U:aQ[ XU5Sn XX-- R U5pXX-- R U5pU R5U:aMQXpUR5U:d[XU5S:wagURnUS:waQURX5nU RU5R U5n URU5R U5nUR5nU"U 5U"U5- $)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)cr"rIrrMrDr0s0r1s1r^r~rlcr&fields r!_rational_function_reconstructionrs4J 66D [[F  A QA  A    ))+/ba #36k//2B36k//2B ))+/ qxxzA~q)Q. B Qw b$ LL  , ,Q / LL  , ,Q / MMOE 8eAh rc@URnUR5H}upgUS:Xa[XqU5nU(d gOXURRnUR U5R5Hup[XU5n U (d gXU 'M! XU'M U$)au 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)rrYrrdrop_to_ground) rKr"rIrr`rLrbr5coeffhmonrrGs r$_rational_reconstruction_func_coeffsrs\ A   66uCF[[%%F..q1;;=6qQ? s >%#'& HrcURn[UR5UR5X#R5upEnUR U5UR U5UR U54$)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 rrr])rrr"rstrLs r _gf_gcdexrLsW 66Dqzz|QZZ\1kkBGA! ??1 tq14??13E EErcURnURU5nURU5nURU5R X/5RU5$)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` )rr_ ground_newrr$)rminpolyr"rp_s r_truncrZsR0 66Dt$G  B >>!  ' / < >t DF !*g ))rcURnURURSURS4S9nURU5nUnUR 5nUR 5nUR S5n [ U5RU5n UR n U(GaXx:Ga[ U5RU5n Xj-URXx- S45U -- nU(aURU5nUR S5nU(aXy:a[ URU55RU5n URU 5URSXy- 45U -- nU(aURU5nUR S5nU(aXy:aMUR 5nU(aXx:aGMU$)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*rrV) rr\rWr_rrnrrrr) rrLrr"rzxringr$r'rdegmlchlcmlcrems r_trial_divisionrsxB 66D ZZa$,,q/ BZ CFt$G C ZZ\F 88:D >>! D a&//$ C **C &.C!!$'g V]A$67== ""1%CAfn V,-66t>!A#q)A-2D>>!A#q)66q9Q>D  !%1a0#GqS!4 ::xQ0 1Q 6  a1a ( a1a (""b(A 6 : FAu  7Iiik]aS!A#Y & q5 "  u8r!u$E"QR&M)A"XXBqrF!/s $ 6 %%'++A ##,,.22A FFKKNz&9KAr4rz##A&r"a!e  : NN1 ! R b Q &lGTQ3RV W 1Aq%1 E 9  6,,$$C((,,Chh))&AUAUAW3::+'(( ,,%%++C((,,C))+A((--fS\\^QWWEUEUEWszz/+,C,( s//23 c"**, - : :1 =q!W00Aw9Z9ZHu &x rcfUS:aX- nXRpCXRpe[US- 5n[U5U:a#X5-nXSX-- pSXdX-- pd[U5U:aM#[ [U55U:agUS:aU*U*pO US:aXVpOgUR 5n U "U 5U "U 5- $)ap Reconstruct a rational number `\frac a b` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are integers. The algorithm is based on the Euclidean Algorithm. In general, `m` is not a prime number, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : Integer `c = \frac a b \; \mathrm{mod} \, m` m : Integer modulus, not necessarily prime domain : IntegerRing `a, b, c` are elements of ``domain`` Returns ======= frac : Rational either `\frac a b` in `\mathbb Q` or ``None`` References ========== 1. [Wang81]_ rrN)rrrintabs get_field) rrIrrrrrrHr^r~rrs r _integer_rational_reconstructionrsH 1u      QKE b'U h#&[B#&[B b'U   3r7|u AvsRC1 a1    E 8eAh rc,URn[UR[5(a[nURR nO[ nUR RnUR5HupgU"XqU5nU(d gXU'M U$)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)rrrr#_rational_reconstruction_int_coeffsrrrY) rKrIrrLreconstructionrrbr5rs rrrswR A$++~..<!!9  &1% ' HrcURnURn[U[5(aMURnURR URR 5S9nUR US9nO)SnUR URR 5S9nUR5upUR5upURU5URU5-n URn Sn /n /n/n[U 5n U RU 5S:XaM#US:XaX-S:HnOU RU 5S:HnU(aMNURU 5nURU 5nURU 5n[UUUU 5nUcMUS:Xa UR$UR5/S/U--nUS:aUUR5HAunnUSUS:XdMUR [#USS5:dM2UR USS&MC UnU n[%XU5H"unnnUU:XdM['UUUU5nUU-nM$ U R)U 5 UR)U5 UR)U5 [+UUU5nUcGMUS:XaUR-5SnOkURRnUR/5H0nURR1UUR-5S5nM2 UR3U5nUR5U5nUR5Sn[7UR3U5UU5(d#[7UR3U 5UU5(dU$GM)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)rrrrrXr\rr@rrrrrrrrYrrrrrr clear_denomsrrrr_r)rrrrrr`QQdomainQQringrErFr,r}r"primeshplistrrr r!minpolypr-rrbr5rKrIr8r7LMhqrLr s r_func_field_modgcd_mr's#D 66D [[F&.)) LL;;$$FMM,C,C,E$F8, 4;;#8#8#:; KKMEB KKMEB KKNT[[^ +E JJE A F F F  aL   a A %  6IND&&q)Q.D   ^^A  ^^A ''* !"b(A 6 :  788Oiik]aSU " q5 "  u8r!u$E"QR&M)A"XXBqrF!/ vv6KAr4rzCBAqQQ7  a b b 0Q ? :  6!!$A--##Cmm''U-?-?-A!-DE) c"A JJt  KKM!  R 0!W== ALL,a 9 9H rc0URn[UR[5(aURRnO URnURnUR 5H>nUR 5H'nU(dM URXFR5nM) M@ UR5HupuUR 5nURRn[UR[5(aURUSS5n[U5n [U 5H]n XZ(dMURXZU-5U-nUSX- S- 4U;aXbUSX- S- 4'MGX'SX- S- 4==U- ss'M_ M U$)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)rrrrrrto_listr denominatorrYrlenr4convert) rrf_rr r5rrbrIr9r<s r _to_ZZ_polyr#sP2 B$++~..## **CAqjjmm4!    KKOO dkk> 2 2 E!"I&A JqAxxNN58c>2Q6!Hac!e$B.,-a!#a%()a!#a%()Q.)& Irc$URnURn[URR[5(avUR 5H`upEUR 5HGupgUS4U-nU"URU5/S/US--5n X;aXU'M;X8==U - ss'MI Mb U$UR 5HDupEUS4nU"URU5/S/US--5n X;aXU'M8X8==U - ss'MF U$)a. Convert a polynomial `f \in \mathbb Z[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha}(z))[x_0]` to a polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` ring : PolyRing `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= f_ : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` rr*)rrrrrrY) rrrr"rbr5rcoefrIrs r _to_ANP_polyr&s0[[F B!&&--00KKMLE"__. 1XK#%FMM$/0A3uQx<?@;qEEQJE/*( IKKMLEq A e,-E!H <=A{1 * IrcvURnUR5Hup4URU5X#'M U$)zc Change representation of the minimal polynomial from ``DMP`` to ``PolyElement`` for a given ring. )rtermsr)rrminpoly_rbr5s r_minpoly_from_denser*0s6 yyH  ++e,( OrcURnUR"[SUR56nURRnU"UR 55nUR nUR5H%n[XV5SnXSR:XdM"XP4s $ XPRURU554$)z Compute the content in `x_0` and the primitive part of a polynomial `f` in `\mathbb Q(\alpha)[x_0, x_1, \ldots, x_{n-1}] \cong \mathbb Q(\alpha)[x_1, \ldots, x_{n-1}][x_0]`. r*r) rrr4rXrrrrfunc_field_modgcdrr^r_)rfringrr#r"rcr5s r_primitive_in_x0r.=s FFE   q%++!6 7D ++  C aiik B 88D -a0 77?7N! t}}U+, ,,rc6URnURnURnX!R:XaUR(de[ X5nUbU$[ S5nUR URU4-URR5S9nUS:Xaf[X5n[X5n URS5RURR55n [XU 5n [X5n O[!U5up[!U5up[#X5SnUR$"['SU56n[X5n[X5n [)URURS55n [XU 5n [X5n [!U 5unn XR+U5-n X R+U5-nXR+U5-nU R-U R.5n XR1U 5UR1U 54$)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 Reconstruction. 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]_ z)rWrr*r)rrrX is_Algebraicrrr\rWrr#rr]modrrr&r.r,rr4r*r_rArr^)rrrrr9rDr0ZZringr"g_rrLcontx0fcontx0gcontx0hZZring_contx0h_s rr,r,RsP 66D [[F A 66>f111 1 ! F   c A ZZ t 3FMM,>,@A  1  !&a( %a( #G5a8''q!5  $  $%fjj',,q/B  1  !&q) !  d ##  d ##  d ## QTTA eeAha  r)F)N)3sympy.core.symbolr sympy.ntheoryrsympy.ntheory.modularrsympy.polys.domainsrsympy.polys.galoistoolsrr r r r sympy.polys.polyerrorsr mpmathrrrr(r/r=rSrfrjrnrrrrrrrrrrrrrrrrrrrrr#r&r*r.r,rrrBs##%.443 ,?.B>B~B:.z-`2j H5V`F>BQhVrPf@?DC L F@>5*pBJ cL=@: zYx7t0f -*T!r