\hMlSrSSKJrJrJrJrJrJrJrJ r J r J r SSK J r JrJrJrJrJrJrJrJrJrJrJr SSKJrJrJrJrJrJrJ r J!r!J"r" SSK#J$r$J%r%J&r&J'r'J(r(J)r) SSK*J+r+J,r, SSK-J.r.J/r/ Sr0S r1S r2S r3S r4S r5Sr6Sr7Sr8Sr9Sr:Sr;SSjrSSjr?SSjr@SSjrASrBSrCg)z8Square-free decomposition algorithms and related tools. ) dup_negdmp_negdup_subdmp_subdup_muldmp_muldup_quodmp_quodup_mul_grounddmp_mul_ground) dup_stripdup_LC dmp_ground_LC dmp_zero_p dmp_ground dup_degree dmp_degree dmp_degree_indmp_degree_list dmp_raise dmp_inject dup_convert) dup_diffdmp_diff dmp_diff_in dup_shift dmp_shift dup_monicdmp_ground_monic dup_primitivedmp_ground_primitive) dup_inner_gcd dmp_inner_gcddup_gcddmp_gcd dmp_resultant dmp_primitive) gf_sqf_list gf_sqf_part)MultivariatePolynomialError DomainErrorcJ[SU55nU[U5:Xdeg)z=Sanity check the degrees of a computed factorization in K[x].c3B# UHupU[U5-v M g7f)N)r).0facks O/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/sqfreetools.py %_dup_check_degrees..$s 9&hsa*S/!&sN)sumr)fresultdegs r1_dup_check_degreesr8"s$ 9& 9 9C *Q-  cS/US--nUH4upE[XA5n[X65VVs/sH upxXuU--PM nnnM6 [U5[X5:Xdegs snnf)z=Sanity check the degrees of a computed factorization in K[X].rN)rziptuple) r5ur6degsr/r0degs_facd1d2s r1_dmp_check_degreesrC(sj 3!a%=D"3**-d*=>*=V *=> ;/!/ // /?sAc ^U(dg[[U[USU5U55(+$)z Return ``True`` if ``f`` is a square-free polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqf_p(x**2 - 2*x + 1) False >>> R.dup_sqf_p(x**2 - 1) True Tr;)rr$r)r5Ks r1 dup_sqf_prF1s* ga!Q):A>???r9c[X5(ag[US-5HAn[USX1U5n[XA5(aM#[XX5n[ XSU5S:wdMA g g)z Return ``True`` if ``f`` is a square-free polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqf_p(x**2 + 2*x*y + y**2) False >>> R.dmp_sqf_p(x**2 + y**2) True Tr;rF)rrangerr%r)r5r>rEifpgcds r1 dmp_sqf_prLGse ! 1Q3Z AqQ ' b   aQ"  #q ( r9c`UR(d [S5eS[URR 5SSUR 5p2[ USUSS9upE[X4SUR 5n[XaR 5(aO[XR*U5US-p M]X U4$)a Find a shift of `f` in `K[x]` that has square-free norm. The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). Returns `(s,g,r)`, such that `g(x)=f(x-sa)`, `r(x)=\text{Norm}(g(x))` and `r` is a square-free polynomial over `k`. Examples ======== We first create the algebraic number field `K=k(a)=\mathbb{Q}(\sqrt{3})` and rings `K[x]` and `k[x]`: >>> from sympy.polys import ring, QQ >>> from sympy import sqrt >>> K = QQ.algebraic_field(sqrt(3)) >>> R, x = ring("x", K) >>> _, X = ring("x", QQ) We can now find a square free norm for a shift of `f`: >>> f = x**2 - 1 >>> s, g, r = R.dup_sqf_norm(f) The choice of shift `s` is arbitrary and the particular values returned for `g` and `r` are determined by `s`. >>> s == 1 True >>> g == x**2 - 2*sqrt(3)*x + 2 True >>> r == X**4 - 8*X**2 + 4 True The invariants are: >>> g == f.shift(-s*K.unit) True >>> g.norm() == r True >>> r.is_squarefree True Explanation =========== This is part of Trager's algorithm for factorizing polynomials over algebraic number fields. In particular this function is algorithm ``sqfr_norm`` from [Trager76]_. See Also ======== dmp_sqf_norm: Analogous function for multivariate polynomials over ``k(a)``. dmp_norm: Computes the norm of `f` directly without any shift. dup_ext_factor: Function implementing Trager's algorithm that uses this. sympy.polys.polytools.sqf_norm: High-level interface for using this function. ground domain must be algebraicrr;Tfront) is_Algebraicr+rmodto_listdomrr&rFrunit)r5rEsgh_rs r1 dup_sqf_normr[isB >>;<< i Aquu5q !Q. !155 ) Q   Q+QUq  7Nr9c ## US-nS/U-nX@4v URn[X5n[[U[ US-555VVs/sHupxUS:dM UPM n nnU H=nUR 5n SX'U V s/sHo*U -PM n n [ X X5n X4v M? SnUS- nU/U-nU*U-/U-n[ UUX5nUU4v M*s snnfs sn f7f)z9Generate a sequence of candidate shifts for dmp_sqf_norm.r;r)rUrsortedr<rHcopyr)r5r>rEns0addirI var_indicess1s1ia1f1jsjajfjs r1_dmp_sqf_norm_shiftsrms  AA qB %K A A"(Qac );"<G"<Q1">> from sympy.polys import ring, QQ >>> from sympy import I >>> K = QQ.algebraic_field(I) >>> R, x, y = ring("x,y", K) >>> _, X, Y = ring("x,y", QQ) We can now find a square free norm for a shift of `f`: >>> f = x*y + y**2 >>> s, g, r = R.dmp_sqf_norm(f) The choice of shifts ``s`` is arbitrary and the particular values returned for ``g`` and ``r`` are determined by ``s``. >>> s [0, 1] >>> g == x*y - I*x + y**2 - 2*I*y - 1 True >>> r == X**2*Y**2 + X**2 + 2*X*Y**3 + 2*X*Y + Y**4 + 2*Y**2 + 1 True The required invariants are: >>> g == f.shift_list([-si*K.unit for si in s]) True >>> g.norm() == r True >>> r.is_squarefree True Explanation =========== This is part of Trager's algorithm for factorizing polynomials over algebraic number fields. In particular this function is a multivariate generalization of algorithm ``sqfr_norm`` from [Trager76]_. See Also ======== dup_sqf_norm: Analogous function for univariate polynomials over ``k(a)``. dmp_norm: Computes the norm of `f` directly without any shift. dmp_ext_factor: Function implementing Trager's algorithm that uses this. sympy.polys.polytools.sqf_norm: High-level interface for using this function. rNr;rTrO) r[rQr+rrRrSrTrmrr&rL)r5r>rErVrWrZrXrYs r1 dmp_sqf_normrosD q$asAy >>;<<!%%--/1q5!QUU3A$Q1-!. !Aquu - Q155 ! !  . a7Nr9cUR(d [S5e[URR 5US-SUR 5n[ XUSS9upE[X4US-UR 5$)ai Norm of ``f`` in ``K[X]``, often not square-free. The domain `K` must be an algebraic number field `k(a)` (see :ref:`QQ(a)`). Examples ======== We first define the algebraic number field `K = k(a) = \mathbb{Q}(\sqrt{2})`: >>> from sympy import QQ, sqrt >>> from sympy.polys.sqfreetools import dmp_norm >>> k = QQ >>> K = k.algebraic_field(sqrt(2)) We can now compute the norm of a polynomial `p` in `K[x,y]`: >>> p = [[K(1)], [K(1),K.unit]] # x + y + sqrt(2) >>> N = [[k(1)], [k(2),k(0)], [k(1),k(0),k(-2)]] # x**2 + 2*x*y + y**2 - 2 >>> dmp_norm(p, 1, K) == N True In higher level functions that is: >>> from sympy import expand, roots, minpoly >>> from sympy.abc import x, y >>> from math import prod >>> a = sqrt(2) >>> e = (x + y + a) >>> e.as_poly([x, y], extension=a).norm() Poly(x**2 + 2*x*y + y**2 - 2, x, y, domain='QQ') This is equal to the product of the expressions `x + y + a_i` where the `a_i` are the conjugates of `a`: >>> pa = minpoly(a) >>> pa _x**2 - 2 >>> rs = roots(pa, multiple=True) >>> rs [sqrt(2), -sqrt(2)] >>> n = prod(e.subs(a, r) for r in rs) >>> n (x + y - sqrt(2))*(x + y + sqrt(2)) >>> expand(n) x**2 + 2*x*y + y**2 - 2 Explanation =========== Given an algebraic number field `K = k(a)` any element `b` of `K` can be represented as polynomial function `b=g(a)` where `g` is in `k[x]`. If the minimal polynomial of `a` over `k` is `p_a` then the roots `a_1`, `a_2`, `\cdots` of `p_a(x)` are the conjugates of `a`. The norm of `b` is the product `g(a1) \times g(a2) \times \cdots` and is an element of `k`. As in [Trager76]_ we extend this norm to multivariate polynomials over `K`. If `b(x)` is a polynomial in `k(a)[X]` then we can think of `b` as being alternately a function `g_X(a)` where `g_X` is an element of `k[X][y]` i.e. a polynomial function with coefficients that are elements of `k[X]`. Then the norm of `b` is the product `g_X(a1) \times g_X(a2) \times \cdots` and will be an element of `k[X]`. See Also ======== dmp_sqf_norm: Compute a shift of `f` so that the `\text{Norm}(f)` is square-free. sympy.polys.polytools.Poly.norm: Higher-level function that calls this. rNr;rTrO)rQr+rrRrSrTrr&)r5r>rErWrXrYs r1dmp_normrq9saP >>;<<!%%--/1q5!QUU3A aAT *DA q1uaee ,,r9c[XUR5n[XRUR5n[X!RU5$)z3Compute square-free part of ``f`` in ``GF(p)[x]``. )rrTr)rR)r5rErWs r1dup_gf_sqf_partrss7A!%% AAuuaee$A q%% ##r9c[S5e)z3Compute square-free part of ``f`` in ``GF(p)[X]``. +multivariate polynomials over finite fieldsNotImplementedErrorr5r>rEs r1dmp_gf_sqf_partry K LLr9c<UR(a [X5$U(dU$UR[X55(a [ X5n[ U[ USU5U5n[XU5nUR(a [X15$[X15S$)z Returns square-free part of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqf_part(x**3 - 3*x - 2) x**2 - x - 2 See Also ======== sympy.polys.polytools.Poly.sqf_part r;) is_FiniteFieldrs is_negativerrr$rr is_Fieldrr )r5rErKsqfs r1 dup_sqf_partrs$ q$$ }}VA\"" AM !XaA& *C !! Czz  S$Q''r9c U(d [X5$UR(a [XU5$[X5(aU$UR [ XU55(a [ XU5nUn[US-5Hn[U[USXAU5X5nM [XX5nUR(a [XQU5$[XQU5S$)z Returns square-free part of a polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqf_part(x**3 + 2*x**2*y + x*y**2) x**2 + x*y r;)rr|ryrr}rrrHr%rr r~rr!)r5r>rErKrIrs r1 dmp_sqf_partrs A!!qQ''!}}]1+,, A!  C 1Q3Zc;q!Q15q< !! Czz**#CA.q11r9c&Un[XUR5n[XRURUS9upE[ U5H"unup[XRU5U4XV'M$ [ X55 UR XAR5U4$)zrErs r1dmp_gf_sqf_listrrzr9cdUR(a [XUS9$UnUR(a[X5n[ X5nO:[ X5up@UR [X55(a[X5nU*n[U5S::aU/4$/Spe[USU5n[XU5upn [U SU5n [XU5nU(dURX45 O>[XU5upn U(d[U5S:aURX45 US- nMq[X55 XE4$)a Return square-free decomposition of a polynomial in ``K[x]``. Uses Yun's algorithm from [Yun76]_. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 >>> R.dup_sqf_list(f) (2, [(x + 1, 2), (x + 2, 3)]) >>> R.dup_sqf_list(f, all=True) (2, [(1, 1), (x + 1, 2), (x + 2, 3)]) See Also ======== dmp_sqf_list: Corresponding function for multivariate polynomials. sympy.polys.polytools.sqf_list: High-level function for square-free factorization of expressions. sympy.polys.polytools.Poly.sqf_list: Analogous method on :class:`~.Poly`. References ========== [Yun76]_ rrr;)r|rr~rrr r}rrrr"rappendr8) r5rErrrr6rIrXrWpqrbs r1 dup_sqf_listrs D q-- Fzzq  aO & == & & AFE!}byAAAqAA!$GA!  Q1  A!  MM1& ! a(a *Q-!# MM1& ! Q v& =r9c[XUS9up4U(a)USSS:Xa[USSX15nUS4/USS-$[U/5nUS4/U-$)aU Return square-free decomposition of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> f = 2*x**5 + 16*x**4 + 50*x**3 + 76*x**2 + 56*x + 16 >>> R.dup_sqf_list_include(f) [(2, 1), (x + 1, 2), (x + 2, 3)] >>> R.dup_sqf_list_include(f, all=True) [(2, 1), (x + 1, 2), (x + 2, 3)] rrr;N)rr r )r5rErrrrWs r1dup_sqf_list_includerAsl$"!C0NE71:a=A% 71:a=% 3Ax'!"+%% ug Ax'!!r9c`U(d [XUS9$UR(a [XX#S9$UnUR(a[ XU5n[ XU5nO=[ XU5upPUR[ XU55(a[XU5nU*n[X5nUS:aU/4$[XU5upp0nUS:wa}[USX5n [X X5upn Sn [U SX5n[XX5n [X5(aXU 'O0[XX5upn U(d[X5S:aXU 'U S- n M^[XqS- X#S9unnX_-nUH%unn U/nX;a[!XUX5X'M!UX'M' [#U5V s/sH oU U 4PM nn [%XAU5 XX4$s sn f)a Return square-free decomposition of a polynomial in `K[X]`. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**5 + 2*x**4*y + x**3*y**2 >>> R.dmp_sqf_list(f) (1, [(x + y, 2), (x, 3)]) >>> R.dmp_sqf_list(f, all=True) (1, [(1, 1), (x + y, 2), (x, 3)]) Explanation =========== Uses Yun's algorithm for univariate polynomials from [Yun76]_ recursively. The multivariate polynomial is treated as a univariate polynomial in its leading variable. Then Yun's algorithm computes the square-free factorization of the primitive and the content is factored recursively. It would be better to use a dedicated algorithm for multivariate polynomials instead. See Also ======== dup_sqf_list: Corresponding function for univariate polynomials. sympy.polys.polytools.sqf_list: High-level function for square-free factorization of expressions. sympy.polys.polytools.Poly.sqf_list: Analogous method on :class:`~.Poly`. rrr;)rr|rr~rrr!r}rrr'rr#rr dmp_sqf_listrr]rC)r5r>rErrrr7contentr6rXrWrrrIrb coeff_contentresult_contentr/s r1rr]sL Ac**qQ00 FzzaA& Q1 %'a0 ==qQ/ 0 0a AFE Q C QwbyqQ'JG F ax Q1 a+a Aq$Aa#A!q #A!/GA!j&*q FA%1A#q$J!M> E!Qe ; 35FIFI !'-Vn 5nay!nnF 5v&) = 6s F+cU(d [XUS9$[XX#S9upEU(a*USSS:Xa[USSXAU5nUS4/USS-$[XA5nUS4/U-$)a< Return square-free decomposition of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> f = x**5 + 2*x**4*y + x**3*y**2 >>> R.dmp_sqf_list_include(f) [(1, 1), (x + y, 2), (x, 3)] >>> R.dmp_sqf_list_include(f, all=True) [(1, 1), (x + y, 2), (x, 3)] rrr;N)rrr r)r5r>rErrrrWs r1dmp_sqf_list_includers|$ #Ac22!!3NE71:a=A% 71:a=%A 6Ax'!"+%% u Ax'!!r9c ~U(d [S5e[X5n[U5(d/$[U[ XR U5U5n[ X!5n[U5H.unupV[U[ XQ"U5*U5U5nXVS-4X4'M0 [XU5n[U5(dU$US4/U-$)z Compute greatest factorial factorization of ``f`` in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_gff_list(x**5 + 2*x**4 - x**3 - 2*x**2) [(x, 1), (x + 2, 4)] zDgreatest factorial factorization doesn't exist for a zero polynomialr;) ValueErrorrrr$rone dup_gff_listrrr )r5rErWHrIrXr0s r1rrs _``!A a== AyEE1-q 1  "1IAv9Q1q115A1u:AD& A! !}}HF8a< r9c<U(d [X5$[U5e)z Compute greatest factorial factorization of ``f`` in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) )rr*rxs r1 dmp_gff_listr s A!!)!,,r9N)F)D__doc__sympy.polys.densearithrrrrrrr r r r sympy.polys.densebasicr rrrrrrrrrrrsympy.polys.densetoolsrrrrrrrr r!sympy.polys.euclidtoolsr"r#r$r%r&r'sympy.polys.galoistoolsr(r)sympy.polys.polyerrorsr*r+r8rCrFrLr[rmrorqrsryrrrrrrrrrrr9r1rs>$$$ ))) ""  0@,DOd%PSlN-b$M !(H"2J , M JZ"8iX">" J-r9