+ iMnRt^RIHtHtHtHtHtHtHtH t H t H t ^RI H t HtHtHtHtHtHtHtHtHtHtHt^RIHtHtHtHtHtHtH t H!t!H"t"^RI#H$t$H%t%H&t&H't'H(t(H)t)^RI*H+t+H,t,^RI-H.t.H/t/Rt0Rt1R t2R t3R t4R t5R t6Rt7Rt8Rt9Rt:Rt;RRltRRlt?RRlt@RRltARtBRtCR#)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 DomainErrorcN\RV44pV\V48XgQhR#)z=Sanity check the degrees of a computed factorization in K[x].c3J"TFwrV\V4,xK R#5i)N)r).0facks& w/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/sqfreetools.py %_dup_check_degrees..$s 9&hsa*S/!!&s!#N)sumr)fresultdegs&& r0_dup_check_degreesr7"s$ 9& 9 9C *Q-  c^.V^,,pVF;wrE\WA4p\W64UUu.uFwrxWuV,,NK pppK= \V4\W48XgQhR#uuppi)z=Sanity check the degrees of a computed factorization in K[X].N)rziptuple) r4ur5degsr.r/degs_facd1d2s &&& r0_dmp_check_degreesrA(sl 3!a%=D"3**-d*=>*=V *=> ;/!/ // /?sA0c bV'gR#\\V\V^V4V44'*#)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 T)rr#r)r4Ks&&r0 dup_sqf_prD1s* ga!Q):A>???r8c\W4'dR#\V^,4FDp\V^W1V4p\WA4'dK$\WW4p\ WSV4^8wgKCR# R#)z Return ``True`` if ``f`` is a square-free polynomial in ``K[X]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x,y = ring("x,y", ZZ) >>> R.dmp_sqf_p(x**2 + 2*x*y + y**2) False >>> R.dmp_sqf_p(x**2 + y**2) True TF)rrangerr$r)r4r<rCifpgcds&&& r0 dmp_sqf_prJGse ! 1Q3Z AqQ ' b   aQ"  #q ( r8crVP'g \R4h^\VPP 4^^VP 4r2\ V^VRR7wrE\W4^VP 4p\WaP 4'dM!\WP)V4V^,r KfW V3#)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 algebraicTfront) is_Algebraicr*rmodto_listdomrr%rDrunit)r4rCsgh_rs&& r0 dup_sqf_normrYisB >>>;<< i Aquu5q !Q. !155 ) Q   Q+QUq 7Nr8c #"V^,p^.V,pW@3xVPp\W4p\\V\ V^,444UUu.uFwrxV^8gKVNK p ppV F@pVP 4p ^W&V U u.uF q)V ,NK p p \ W W4p W3xKB ^pV^, pV.V,pV)V,.V,p\ VVW4pVV3xK;uuppiuup i5i)z9Generate a sequence of candidate shifts for dmp_sqf_norm.)rSrsortedr:rFcopyr)r4r<rCns0addirG var_indicess1s1ia1f1jsjajfjs&&& r0_dmp_sqf_norm_shiftsrks  AA qB %K A A"(Qac );"<G"<Q11">> 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. rLTrM) rYrOr*rrPrQrRrkrr%rJ)r4r<rCrTrUrXrVrWs&&& r0 dmp_sqf_normrmsD q$asAy >>>;<<!%%--/1q5!QUU3A$Q1-!. !Aquu - Q155 ! !  . a7Nr8cVP'g \R4h\VPP 4V^,^VP 4p\ WVRR7wrE\W4V^,VP 4#)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. rLTrM)rOr*rrPrQrRrr%)r4r<rCrUrVrWs&&& r0dmp_normro9scP >>>;<<!%%--/1q5!QUU3A aAT *DA q1uaee ,,r8c\WVP4p\WPVP4p\W!PV4#)z3Compute square-free part of ``f`` in ``GF(p)[x]``. )rrRr(rP)r4rCrUs&& r0dup_gf_sqf_partrqs7A!%% AAuuaee$A q%% ##r8c\R4h)z3Compute square-free part of ``f`` in ``GF(p)[X]``. +multivariate polynomials over finite fieldsNotImplementedErrorr4r<rCs&&&r0dmp_gf_sqf_partrw K LLr8cLVP'd \W4#V'gV#VP\W44'd \ W4p\ V\ V^V4V4p\WV4pVP'd \W14#\W14^,#)z Returns square-free part of a polynomial in ``K[x]``. Examples ======== >>> from sympy.polys import ring, ZZ >>> R, x = ring("x", ZZ) >>> R.dup_sqf_part(x**3 - 3*x - 2) x**2 - x - 2 See Also ======== sympy.polys.polytools.Poly.sqf_part ) is_FiniteFieldrq is_negativer rr#rris_Fieldrr)r4rCrIsqfs&& r0 dup_sqf_partr~s$ q$$ }}VA\"" AM !XaA& *C !! Czzz  S$Q''r8c V'g \W4#VP'd \WV4#\W4'dV#VP \ WV44'd \ WV4pTp\V^,4Fp\V\V^WAV4W4pK \WW4pVP'd \WQV4#\WQV4^,#)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~rzrwrr{rrrFr$rr r|rr )r4r<rCrIrGr}s&&& r0 dmp_sqf_partrs A!!qQ''!}}]1+,, A!  C 1Q3Zc;q!Q15q< !! Czzz**#CA.q11r8c,Tp\WVP4p\WPVPVR7wrE\ V4F"wpwr\WPV4V3WV&K$ \ W54VP WAP4V3#)z>> 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]_ r)rzrr|r rrr{rrrr!rappendr7) r4rCrrrr5rGrVrUpqr`s &&& r0 dup_sqf_listrsD q-- Fzzzq  aO & == & & AFE!}byAAAqAA!$GA!  Q1  A!  MM1& ! a(a *Q-!# MM1& ! Qv& =r8c\WVR7wr4V'dCV^,^,^8Xd.\V^,^,W14pV^3.VR,,#\V.4pV^3.V,#)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)] rNN)rr r )r4rCrrrrUs&&& r0dup_sqf_list_includerAsj$"!C0NE71:a=A% 71:a=% 3Ax'"+%% ug Ax'!!r8cV'g\WVR7#VP'd\WW#R7#TpVP'd\ WV4p\ WV4pM>\ WV4wrPVP\ WV44'd\WV4pV)p\W4pV^8dV.3#\WV4wrp/pV^8wd\V^W4p \W W4wrp ^p \V ^W4p\WW4p \W4'dWV &M6\WW4wrp V'g\W4^8dWV &V ^, p Kf\Wq^, W#R7wppW_,pVF*wpp V.pW9d\!W,VW4W&K&VW&K, \#V4U u.uF qV ,V 3NK pp \%WAV4WX3#uup i)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`. r)rrzrr|rrr r{rrr&rr"rr dmp_sqf_listrr[rA)r4r<rCrrrr6contentr5rVrUrrrGr` coeff_contentresult_contentr.s&&&& r0rr]sL Ac**qQ00 FzzzaA& Q1 %'a0 ==qQ/ 0 0a AFE Q C QwbyqQ'JG F ax Q1 a+a Aq$Aa#A!q #A!/GA!j&*q FA$0A#q$J!M> E!Qe ; 35FIFI !'-Vn 5nay!nnF 5v&) = 6s-GcV'g\WVR7#\WW#R7wrEV'dDV^,^,^8Xd/\V^,^,WAV4pV^3.VR,,#\WA4pV^3.V,#)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)] rr)rrr r)r4r<rCrrrrUs&&&& r0dmp_sqf_list_includersz$ #Ac22!!3NE71:a=A% 71:a=%A 6Ax'"+%% u Ax'!!r8c V'g \R4h\W4p\V4'g.#\V\ WP V4V4p\ W!4p\V4F2wpwrV\V\ WQ!V4)V4V4pWV^,3W4&K4 \WV4p\V4'gV#V^3.V,#)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 polynomial) ValueErrorrrr#rone dup_gff_listrrr)r4rCrUHrGrVr/s&& r0rrs _``!A a== AyEE1-q 1  "1IAv9Q1q115A1u:AD& A! !}}HF8a< r8c>V'g \W4#\V4h)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)rvs&&&r0 dmp_gff_listr s A!!)!,,r8N)F)D__doc__sympy.polys.densearithrrrrrrrr r r sympy.polys.densebasicr r rrrrrrrrrrsympy.polys.densetoolsrrrrrrrrr sympy.polys.euclidtoolsr!r"r#r$r%r&sympy.polys.galoistoolsr'r(sympy.polys.polyerrorsr)r*r7rArDrJrYrkrmrorqrwr~rrrrrrrrrr8r0rs>$$$ ))) ""  0@,DOd%PSlN-b$M !(H"2J , M JZ"8iX">" J-r8