+ ii:^RIHt^RIHt^RIHt^RIHt^RIH t H t H t H t H t ^RIHt^RIHt^RItR tR tR tR tR tRtRtRtRtRtRtR)RltRtRt Rt!Rt"Rt#Rt$Rt%Rt&Rt'R*Rlt(Rt)R t*R!t+R"t,R#t-R$t.R%t/R&t0R't1R(t2R#)+)Dummy) nextprimecrt)PolynomialRing)gf_gcd gf_from_dictgf_gcdexgf_divgf_lcm)ModularGCDFailed)sqrtNcVPpV'g,V'g$VPVPVP3#V'gYVPVPP8dV)VPVP)3#WPVP3#V'gYVPVPP8dV)VP)VP3#WPVP3#R#)zb Compute the GCD of two polynomials in trivial cases, i.e. when one or both polynomials are zero. N)ringzeroLCdomainone)fgrs&& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/modulargcd.py _trivial_gcdr s 66D yy$))TYY..  44$++"" "2tyy488)+ +ii) )  44$++"" "2y$))+ +hh ) ) cVPPpV'dTpVP4pVPVPV4pVP4pWu8dMNWAP Wu, 34P WdP,4, PV4pKfTpTpKVP VPVPV44PV4#)zE Compute the GCD of two univariate polynomials in `\mathbb{Z}_p[x]`. )rrdegreeinvertr mul_monom mul_ground trunc_ground)fpgppdomremdeglcinvdegrems&&& r_gf_gcdr(#s ''..C iik 255!$ZZ\F|v|o6AA%&&.QQ__`abC   ==BEE1- . ; ;A >>rcDVPPPVPVP4p^p\ V4pW#,^8Xd\ V4pKVP V4pVP V4p\ WEV4pVP4pV#)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 )rrgcdrrrr(r)rrgammar"r r!hpdeghps&& r_degree_bound_univariater.:s~& FFMM  addADD )E A! A )q. aL  B  B  B IIKE Lrc hVP4pVPP^,pVPPp\ V^,4FHp\ W#.VP WW,4VP WW,4.RR7^,Wg3&KJ VP4V#)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 T symmetric)rrgensrrangercoeff strip_zero)r,hqr"qnxhpqis&&&& r,_chinese_remainder_reconstruction_univariater<[sl A  QA '',,C 1Q3Z!$!$ @DQRSTD NN Jrc*VPVP8Xd'VPPP'gQh\W4pVeV#VPpVP 4wr@VP 4wrQVPP WE4p\ W4pV^8Xd6V!V4VPWF,4VPWV,43#VPP VPVP4p^p ^p \V 4p W,^8Xd\V 4p KVPV 4p VPV 4p \WV 4p V P4pW8dKlW8d^p TpKxV PV4PV 4p V ^8XdT p T pK\V XW4pW,p VV8XgTpKVPVP44pVP!V4wppVP!V4wppV'dEKV'dEK"VP^8dV)pVPV4pVPWF,4pVPWV,4pVVV3#)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]_ )rris_ZZr primitiver*r.rrrrr(rr< quo_groundcontentdiv)rrresultrcfcgchboundr+mr"r r!r,r-hlastmhmhfquofremgquogremcffcfgs&& rmodgcd_univariaterRsF 66QVV  3 3 33 3 ! F  66D KKMEB KKMEB  B $Q *E zBxbh/bh1GGG KKOOADD!$$ 'E A A  aLi1n! A ^^A  ^^A  RQ   =  ]AE  ]]5 ! . .q 1 6AF  9"fa K V|F  MM"**, 'UU1X dUU1X dtDDttaxS R A//"(+C//"(+Cc3; rc VPpVPpVPp/pVP4F+wrgVRRV9d/WVRR&WuVRR,VR,&K- .p\ VP 44Fp\ V\WqV4W4pK VPVPV^, ,R7p V PV4PV4p WPV PV443#)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) Nsymbols)rrngens itertermsitervaluesrr clonerU from_denserquoset_ring) rr"rr#kcoeffsmonomr4contyringcontfs && r _primitiveresR 66D ++C A F  ":V #!#F": (-uSbz59%& Dfmmo&dL37@' JJt||AaC0J 1E   T " / / 2E %%t,- --rcVPPpRV^, ,pVP4FpVRRV8gKVRRpK V#)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) N)rrV)rrW itermonoms)rr_degfras& r_degriZsOP  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 rTNrV)rrWr[rUr2rirrX) rrr_rcyrhlcfrar4s & r_LCrmsP 66D A JJt||AaC0J 1E 1 A 7D **C  ":  "I% %C& JrcVPpVPpVP4F,wrEWA,3VRV,WA^,R,pWSV&K. V#)zK Make the variable `x_i` the leading one in a multivariate polynomial `f`. N)rrrX)rr;rfswaprar4 monomswaps&& r_swaprqsX 66D IIE XK%)+eaCDk9  i& LrcVPpVPPVPVP4pVPP\ V^4P\ V^4P4pW4,p^p\ V4pWV,^8Xd\ V4pKVP V4pVP V4p\Wv4wr\W4wr\WV4p V P4p \\V4\V4V4p \V4FpV P^V4V,'gK$VP^V4P V4pVP^V4P V4p\VVV4pVP4pVV 3u# \VP4VP44V 3#)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]_ )rrr*rrqrrrer(rrmr3evaluatemin)rrrgamma1gamma2 badprimesr"r r!contfpcontgpconthp ycontbounddeltaafpagpahpaxbounds&& r_degree_bound_bivariaters~T 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 c><S!\W#.W.RR7^,4#)Tr0r)cpcqr"r7rs&&&&rcrt_<_chinese_remainder_reconstruction_multivariate..crt_ls #qfrh$?BC Cr) setmonoms intersectiondifference_updaterrr isinstancer._chinese_remainder_reconstruction_multivariate) r,r6r"r7hpmonomshqmonomsrrr:rrars &&&& @rrrsP299;H299;H  " "8 ,F v& v& WW^^F ;;D '',,C"''...11= D")RY5 ")Ta0 $5 10  Jrc(VPpV'd5VPPpVPPV,pMVPpVPV,p\W4FwrVPp VPp VF)p W8XdK WV , ,p WV , ,p K+ VP W4p V P V 4pWjPV4V,, pK VPV4#)a Reconstruct a polynomial `h_p` in `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` from a list of evaluation points in `\mathbb{Z}_p` and a list of polynomials in `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, which are the images of `h_p` evaluated in the variable `x_i`. It is also possible to reconstruct a parameter of the ground domain, i.e. if `h_p` is a polynomial over `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]`. In this case, one has to set ``ground=True``. Parameters ========== evalpoints : list of Integer objects list of evaluation points in `\mathbb{Z}_p` hpeval : list of PolyElement objects list of polynomials in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{k-1}]`, images of `h_p` evaluated in the variable `x_i` ring : PolyRing `h_p` will be an element of this ring i : Integer index of the variable which has to be reconstructed p : Integer prime number, modulus of `h_p` ground : Boolean indicates whether `x_i` is in the ground domain, default is ``False`` Returns ======= hp : PolyElement interpolated polynomial in (resp. over) `\mathbb{Z}_p[x_0, \ldots, x_{k-1}]` ) rrr2ziprrrr^r) evalpointshpevalrr;r"groundr,rrkr}rnumerdenombr4s&&&&&& r_interpolate_multivariaterysN B ## KK  Q  IIaLj) Av UNE UNE  e'  ' ll4 5((* ??1 rc& VPVP8Xd'VPPP'gQh\W4pVeV#VPpVP 4wr@VP 4wrQVPP WE4p\ W4wrxYxu;8Xd^8Xd9MM5V!V4VPWF,4VPWV,43#\V^4p \V^4p V P4p V P4p \ W4wrYu;8Xd^8Xd9MM5V!V4VPWF,4VPWV,43#VPP VPVP4pVPP V PV P4pVV,p^p^p\V4pVV,^8Xd\V4pKVPV4pVPV4p\VV4wpp\VV4wpp\VVV4pVP4pVV8dKVV8d^pTpK\\V4\V4V4pVP4pVP4pVP4p\!V V, V V, W, V,4^,pVV8dEK"^p.p .p!Rp"\#V4Fp#VP%^V#4p$V$V,'gK&VP%^V#4PV4p%VP%^V#4PV4p&\V%V&V4p'V'P4p(V(V8dKV(V8d ^pT(pRp"MXV'PV$4PV4p'V P'V#4V!P'V'4V^, pVV8XgKM V"'dEK5VV8dEK?\)V V!V^V4p)\V)V4^,p)V)VP+V4,p)V)P^4p*V*V 8dEKV*V 8d^pT*p EKV)PV4PV4p)V^8XdTpT)p+EK\-V)X+VV4p,VV,pV,V+8XgT,p+EKV,P/V,P144p-VP3V-4wp.p/VP3V-4wp0p1V/'dEKEV1'dEKPV-P^8dV)pV-PV4p-V.PWF,4p2V0PWV,4p3V-V2V33#)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]_ TF)rrr>rr?r*rrrqrrrrrer(rmrtr3rsappendrr^rr@rArB)4rrrCrrDrErFrr{rogswapdegyfdegygybound xcontboundrurvrwrHr"r r!rxryrz degconthpr| degcontfp degcontgpdegdeltaNr8rrunluckyr}deltaar~rrdeghpar,degyhprIrJrKrLrMrNrOrPrQs4&& rmodgcd_bivariatersR 66QVV  3 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 ^^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 dtDDttaxS R A//"(+C//"(+Cc3; rc \VPpVPpV^8XdT\WV4PV4pVP 4pW^,8dR#W^,8d W^&\ hV#VP V^, 4p VP V^, 4p \ W4wr\ W4wr\WV4p V P 4pV P 4pV P 4pVWF^, ,8dR#VWF^, ,8dVWF^, &\ h\V4p\V4p\VVV4pTp\V^, 4F=pV\\\VV44\\VV44V4,pK? VP 4p\W, W, W6^, ,WF^, ,, V,4^,pVV8dR#^p^p.p.p\\V44pV'Ed\P!V^4^,pVPV4VP^V4V,'gKYVP^V4V,pVPV^, V4PV4pVPV^, V4PV4p \!VV W#V4p!V!fV^, pVV8dR#KV!P"'d#V P%V4PV4pV#V!P'V4PV4p!VP)V4VP)V!4V^, pVV8XgEKq\+VVWV^, V4p\ Wr4^,V P%V4,pVP V^, 4p"V"W6^, ,8dR#V"W6^, ,8dV"W6^, &\ hV#R#)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]_ N)rrWr(rrr rermr3rqrtlistrandomsampleremovers_modgcd_multivariate_p is_groundr^rrr)#rrr"degbound contboundrr_rKdeghrrrdcontgconthdegcontfdegcontgdegconthrllcgr|evaltestr;rrr8drhevalpointsr}rfagahadegyhs#&&&&& rrrst` 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 &&H rc &VPVP8Xd'VPPP'gQh\W4pVeV#VPpVPpVP 4wrPVP 4wraVPP WV4pVPP VPVP4pVPPp \V4FKp WPP \W 4P\W4P4,p KM \VP4VP44U U u.uFwr\W4NK p p p \V 4p^p^p\V4pV V,^8Xd\V4pKVP!V4pVP!V4p\#VVVW4pTfK^TP'T4P!T4pT^8XdTpTpK\)TXTT4pTT,pTT8XgTpKTP 4^,pTP+T4wppTP+T4wppT'dKT'dEKTP^8dT)pTP'T4pTP'YW,4pTP'Yg,4pTTT3#uup p i \$d^pEKpi;i)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 )rrr>rrWr?r*rrr3rqrdegreesrtrrrrr rrrB)rrrCrr_rDrErFr+rwr;fdeggdegrrrHr"r r!r,rIrJrKrLrMrNrOrPrQs&& rmodgcd_multivariater's}\ 66QVV  3 3 33 3 ! F  66D A KKMEB KKMEB  B KKOOADD!$$ 'E I 1X[[__U1[^^U1[^^DD 36aiik199;2OP2OJDD2OHPXI A A  aL!mq ! A ^^A  ^^A  'B8GB :  ]]5 ! . .q 1 6AF  ;B1 M QV|F  LLN1 UU1X dUU1X dtDDttaxS R A//"(+C//"(+Cc3; ]Q"  A  s7K8)K>> LLcVPp\VP4VP4W#P4wrEVP V4VP V43#)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 $dooh&? ??rcVPpVPpVP4pV^,pWV, ^, pY#PrYPrV P4V8dR\ WV4^,p YW,, P V4rYW,, P V4rKfYrVP4V8g\WV4^8wdR#VPpV^8wdRVPW4pV PV4P V4p VPV4P V4pVP4pV!V 4V!V4, #)a Reconstruct a rational function `\frac a b` in `\mathbb Z_p(t)` from .. math:: c = \frac a b \; \mathrm{mod} \, m, where `c` and `m` are polynomials in `\mathbb Z_p[t]` and `m` has positive degree. The algorithm is based on the Euclidean Algorithm. In general, `m` is not irreducible, so it is possible that `b` is not invertible modulo `m`. In that case ``None`` is returned. Parameters ========== c : PolyElement univariate polynomial in `\mathbb Z[t]` p : Integer prime number m : PolyElement modulus, not necessarily irreducible Returns ======= frac : FracElement either `\frac a b` in `\mathbb Z(t)` or ``None`` References ========== 1. [Hoeij04]_ N) rrrrrrrr(rrrto_field)cr"rHrrMrDr0s0r1s1r]r}rlcr&fields&&& r!_rational_function_reconstructionrs#J 66D [[F  A QA  A    ))+/ba #36k//2B36k//2B qxxzA~q)Q. B Qw b$ LL  , ,Q / LL  , ,Q / MMOE 8eAh rcJVPpVP4FwrgV^8Xd\WqV4pV'gR#MZVPPpVP V4P4F!wr\WV4p V 'gR#WV &K# WV&K V#)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 N)rrXrrdrop_to_ground) rJr"rHrr_rKrar4coeffhmonrrFs &&&&& r$_rational_reconstruction_func_coeffsrs\ A   66uCF[[%%F..q1;;=6qQ? s >%#'& HrcVPp\VP4VP4W#P4wrEpVP V4VP V4VP V43#)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"rstrKs&&& r _gf_gcdexrLsW 66Dqzz|QZZ\1kkBGA! ??1 tq14??13E EErcVPpVPV4pVPV4pVPV4P W.4PV4#)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 ))rcVPpVPVP^,VP^,3R7pVPV4pTpVP 4pVP 4pVP ^4p \ V4PV4p VP p V'Ed%Wx8Ed\ V4PV4p Wj,VPWx, ^34V ,, pV'dVPV4pVP ^4pV'dWy8d\ VPV44PV4p VPV 4VP^Wy, 34V ,, pV'dVPV4pVP ^4pKVP 4pEK-V#)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]_ rT) rr[rUr^rrmrrrr) rrKrr"rzxringr$r'rdegmlchlcmlcrems &&&& r_trial_divisionrsdB 66D ZZa$,,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"R&M)A"XXBrF!/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!W00Aw9Z9ZH rcV^8d W, pYPrCYPre\V^, 4p\V4V8d)W5,pYSW,, rSYdW,, rdK8\ \V44V8dR#V^8dV)V)rM V^8dYVrMR#VP 4p V !V 4V !V 4, #)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]_ N)rrrintabs get_field) rrHrrrrrrGr]r}rrs &&& r _integer_rational_reconstructionrsH 1u      QKE b'U h#&[B#&[B 3r7|u AvsRC1 a1    E 8eAh rc2VPp\VP\4'd\pVPP pM\ pVP PpVP4FwrgV!WqV4pV'gR#WV&K V#)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_coeffsrrrX) rJrHrrKreconstructionrrar4rs &&& rrrswR A$++~..<!!9  &1% ' HrcVPpVPp\V\4'dTVPpVPP VPP 4R7pVP VR7pM,^pVP VPP 4R7pVP4wrVP4wrVPV4VPV4,p VPp ^p .p .p.p\V 4p V PV 4^8XdK$V^8Xd W,^8HpMV PV 4^8HpV'dKUVPV 4pVPV 4pVPV 4p\VVVV 4pVfKV^8Xd VP#VP4.^.V,,pV^8ddVP4FOwppV^,V^,8XgKVP \#VR,48gK@VP VR&KQ TpT p\%WV4F'wpppVV8XgK\'VVVV4pVV,pK) V P)V 4VP)V4VP)V4\+VVV4pVfEKV^8XdVP-4^,pMoVPPpVP/4F4pVPP1VVP-4^,4pK6 VP3V4pVP5V4pVP4^,p\7VP3V4VV4'dEK\7VP3V 4VV4'dEKV#)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 rr)rrrrrWr[rr?rrrrrrrrXrrrrrr clear_denomsrrrr^r)rrrrrr_QQdomainQQringrDrEr+r|r"primeshplistrrr r!minpolypr,rrar4rJrHr7r6LMhqrKr s&&& r_func_field_modgcd_mr'sD 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"R&M)A"XXBrF!/ vv6KAr4rzCBAqQQ7  a b b 0Q ? :  6!!$A--##Cmm''U-?-?-A!-DE) c"A JJt  KKM!  R 0!W== ALL,a 9 9HrcVPp\VP\4'dVPPpM VPpVPpVP 4F?pVP 4F(pV'gK VPWFP4pK* KA VP4EFwruVP 4pVPPp\VP\4'dVPVR,4p\V4p \V 4Fp WZ,'gKVPWZ,V,4V,pV^,W, ^, 3V9dWbV^,W, ^, 3&KqW'^,W, ^, 3;;,V, uu&K EK V#)a: Compute an associate of a polynomial `f \in \mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` in `\mathbb Z[x_1, \ldots, x_{n-1}][z] / (\check m_{\alpha}(z))[x_0]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` ring : PolyRing `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` Returns ======= f_ : PolyElement associate of `f` in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` r)rrrrrrto_listr denominatorrXrlenr3convert) rrf_rr r4rrarHr8r;s && r _to_ZZ_polyr#sN2 B$++~..## **CAqjjmm4!    KKOO dkk> 2 2 E"I&A JqAxxNN58c>2Q6!Hac!e$B.,-a!#a%()a!#a%()Q.)& IrcVPpVPp\VPP\4'dVP 4F}wrEVP 4FdwrgV^,3V,pV!VPV4.^.V^,,,4p W9dWV&KPW8;;,V , uu&Kf K V#VP 4F]wrEV^,3pV!VPV4.^.V^,,,4p W9dWV&KIW8;;,V , uu&K_ V#)a. Convert a polynomial `f \in \mathbb Z[x_1, \ldots, x_{n-1}][z]/(\check m_{\alpha}(z))[x_0]` to a polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]`, where `\check m_{\alpha}(z) \in \mathbb Z[z]` is the primitive associate of the minimal polynomial `m_{\alpha}(z)` of `\alpha` over `\mathbb Q`. Parameters ========== f : PolyElement polynomial in `\mathbb Z[x_1, \ldots, x_{n-1}][x_0, z]` ring : PolyRing `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` Returns ======= f_ : PolyElement polynomial in `\mathbb Q(\alpha)[x_0, \ldots, x_{n-1}]` )rrrrrrX) rrrr"rar4rcoefrHrs && r _to_ANP_polyr&s0[[F B!&&--00KKMLE"__. 1XK#%FMM$/0A3uQx<?@;qEEQJE/*( IKKMLEq A e,-E!H <=A{1 * IrcvVPpVP4Fwr4VPV4W#&K V#)zc Change representation of the minimal polynomial from ``DMP`` to ``PolyElement`` for a given ring. )rtermsr)rrminpoly_rar4s&& r_minpoly_from_denser*0s6 yyH  ++e,( OrcVPpVP!\^VP4!pVPPpV!VP 44pVP pVP4F*p\WV4^,pWSP8XgK'WP3u# WPPVPV443#)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]`. ) rrr3rWrrrrfunc_field_modgcdrr]r^)rfringrr#r"rbr4s& r_primitive_in_x0r.=s FFE   q%++!6 7D ++  C aiik B 88D -a0 77?7N! t}}U+, ,,rcnVPpVPpVPpW!P8XdVP'gQh\ W4pVeV#\ R4pVP VPV3,VPP4R7pV^8Xdg\W4p\W4p VP^4PVPP44p \WV 4p \W4p M\!V4wr\!V4wr\#W4^,pVP$!\'^V4!p\W4p\W4p \)VPVP^44p \WV 4p \W4p \!V 4wpp WP+V4,p W P+V4,pWP+V4,pV P-V P.4p WP1V 4VP1V 43#)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)rUr)rrrW is_Algebraicrrr[rUrr#rr\modrrr&r.r,rr3r*r^r@rr])rrrrr8rCr0ZZringr"g_rrKcontx0fcontx0gcontx0hZZring_contx0h_s&& rr,r,RsP 66D [[F A 66>f1111 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<rRrerirmrqrrrrrrrrrrrrrrrrrrr#r&r*r.r,rrrBs##%.443 ,?.B>B~B:.z-`2j H5V`F>BQhVrPf@?DC L F@>5*pBJ cL=@: zYx7t0f -*T!r