+ i;Rt^RIHt^RIHt^RIHtHt^RIH t H t H t ^RI H t ^RIHt^RIHtHt^RIHt^R IHtHt^R IHt^R IHt^R IHt!R R4t!RR4tR#)a This module contains functions for two multivariate resultants. These are: - Dixon's resultant. - Macaulay's resultant. Multivariate resultants are used to identify whether a multivariate system has common roots. That is when the resultant is equal to zero. )prod)Mul)Matrixdiag)Poly degree_listrem)simplify) IndexedBase) itermonomials monomial_deg) monomial_key)poly_from_expr total_degree)binomial)combinations_with_replacement)sympy_deprecation_warningcja]tRt^toRtRt]R4tRtRt Rt Rt Rt R t R tR tR tVtR #)DixonResultanta A class for retrieving the Dixon's resultant of a multivariate system. Examples ======== >>> from sympy import symbols >>> from sympy.polys.multivariate_resultants import DixonResultant >>> x, y = symbols('x, y') >>> p = x + y >>> q = x ** 2 + y ** 3 >>> h = x ** 2 + y >>> dixon = DixonResultant(variables=[x, y], polynomials=[p, q, h]) >>> poly = dixon.get_dixon_polynomial() >>> matrix = dixon.get_dixon_matrix(polynomial=poly) >>> matrix Matrix([ [ 0, 0, -1, 0, -1], [ 0, -1, 0, -1, 0], [-1, 0, 1, 0, 0], [ 0, -1, 0, 0, 1], [-1, 0, 0, 1, 0]]) >>> matrix.det() 0 See Also ======== Notebook in examples: sympy/example/notebooks. References ========== .. [1] [Kapur1994]_ .. [2] [Palancz08]_ c aWnW n\VP4Vn\VP4Vn\ R4p\ VP4Uu.uF qCV,NK upVn\ VP4Uau.uF"o\V3RlVP44NK$ upVn R#uupiuupi)a A class that takes two lists, a list of polynomials and list of variables. Returns the Dixon matrix of the multivariate system. Parameters ---------- polynomials : list of polynomials A list of m n-degree polynomials variables: list A list of all n variables alphac3H<"TFp\V4S,xK R#5i)N)r).0polyis& ڃ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/polys/multivariate_resultants.py *DixonResultant.__init__..[s SBR$T!21!5!5BRs"N) polynomials variableslennmr rangedummy_variablesmax _max_degrees)selfrrars&&& `r__init__DixonResultant.__init__Ds'"T^^$T%%&  .3DFFms 'AA$c  VPV4p\VPV4p\VR\ RVP4R7p\ VP 4UUu.uF9pVUu.uF)p\V.VPO5!PV4NK+ upNK; upp4pVP^,VP^,8wd\VPR,4Uu.uFSp\;QJd%RVRV3,4F 'gK RM RM!RVRV3,44'gKQVNKU ppVRV3,pV#uupiuuppiuupi)z Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. Tlexreversekeyc3*"TF q^8gxK R#5iN)relements& rr2DixonResultant.get_dixon_matrix..s42-4a<2sNNNF) rUr rsortedr rrQrcoeff_monomialshaper#any) r'rRr2 monomialscr" dixon_matrixcolumnkeeps && rget_dixon_matrixDixonResultant.get_dixon_matrixsT**:6 "$..+> 9d+E4>>BD )3):):)<>)?   a L$6$6q$9 9).|/A/A"/E)F5)Fvs4'6 24sss4'6 244F)FD5(40L 4> 5s0" E +/EE 0E#E#!E#E#E c  aaSP'dR#SPwr#\SP4^,4o\ V4Uau.uF[o\ ;QJd)VV3Rl\ V44F 'gK RM RM!VV3Rl\ V444'gKYSNK] ppSVR3,o\ ^.V^, ,^.,.4pSR,V8XdR#R#uupi)aw Test for the validity of the Kapur-Saxena-Yang precondition. The precondition requires that the column corresponding to the monomial 1 = x_1 ^ 0 * x_2 ^ 0 * ... * x_n ^ 0 is not a linear combination of the remaining ones. In SymPy notation this is the last column. For the precondition to hold the last non-zero row of the rref matrix should be of the form [0, 0, ..., 1]. Fc3@<"TFpSSV3,^8gxK R#5ir]r_)rjrmatrixs& rr2DixonResultant.KSY_precondition..s*Oh6!Q$<1+ conditions&f ` rKSY_preconditionDixonResultant.KSY_preconditions  ||&++-*+ 8P8ass*OeAh*Osss*OeAh*O'O8PQQC1IO,- $<9 $QsC6#C6%C6.C6c B\VP4Uu.uF'q!PV4P'dK%VNK) pp\VP4Uu.uF'qAP V4P'dK%VNK) ppWV3,#uupiuupi)z/Remove the zero rows and columns of the matrix.)r#r>rowrtcolscol)r'rrrr>rqr{s&& rdelete_zero_rows_and_columns+DixonResultant.delete_zero_rows_and_columnssV[[)O)!A1M1MAA) OV[[)O)!A1M1MAA) ODj!! OOs"BB"BBc ^p\VP4F-pVPV4FpV^8wgK W$,pK+ K/ V#)z;Calculate the product of the leading entries of the matrix.)r#r>rz)r'rrresrzels&& rproduct_leading_entries&DixonResultant.product_leading_entriessD%Cjjo7(C&&  r4c VPV4pVP4wr#pVP\V44pVPV4#)z@Calculate the Kapur-Saxena-Yang approach to the Dixon Resultant.)r}LUdecompositionr r)r'rr_Us&& rget_KSY_Dixon_resultant&DixonResultant.get_KSY_Dixon_resultantsI226:((*a228A;?++F33r4)r&r$r"r!rrN)__name__ __module__ __qualname____firstlineno____doc__r)propertyr2rHrNrUrmrwr}rr__static_attributes____classdictcell__ __classdict__s@rrrsS(T$4!!"IH& 86"44r4rcTa]tRt^toRtRtRtRtRtRt Rt Rt R t R t VtR #) MacaulayResultanta A class for calculating the Macaulay resultant. Note that the polynomials must be homogenized and their coefficients must be given as symbols. Examples ======== >>> from sympy import symbols >>> from sympy.polys.multivariate_resultants import MacaulayResultant >>> x, y, z = symbols('x, y, z') >>> a_0, a_1, a_2 = symbols('a_0, a_1, a_2') >>> b_0, b_1, b_2 = symbols('b_0, b_1, b_2') >>> c_0, c_1, c_2,c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4') >>> f = a_0 * y - a_1 * x + a_2 * z >>> g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2 >>> h = c_0 * y * z ** 2 - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + c_4 * z ** 3 >>> mac = MacaulayResultant(polynomials=[f, g, h], variables=[x, y, z]) >>> mac.monomial_set [x**4, x**3*y, x**3*z, x**2*y**2, x**2*y*z, x**2*z**2, x*y**3, x*y**2*z, x*y*z**2, x*z**3, y**4, y**3*z, y**2*z**2, y*z**3, z**4] >>> matrix = mac.get_matrix() >>> submatrix = mac.get_submatrix(matrix) >>> submatrix Matrix([ [-a_1, a_0, a_2, 0], [ 0, -a_1, 0, 0], [ 0, 0, -a_1, 0], [ 0, 0, 0, -a_1]]) See Also ======== Notebook in examples: sympy/example/notebooks. References ========== .. [1] [Bruce97]_ .. [2] [Stiller96]_ c HWnW n\V4VnVPUu.uFp\ V.VPO5!NK upVnVP 4VnVP4Vn VPVP4Vn R#uupi)z Parameters ========== variables: list A list of all n variables polynomials : list of SymPy polynomials A list of m n-degree polynomials N) rrr r!rdegrees _get_degree_mdegree_mget_sizemonomials_sizeget_monomials_of_certain_degree monomial_set)r'rrrs&&& rr)MacaulayResultant.__init__+s'"Y ++-+AE T;DNN;+- **, "mmo!@@O-s Bc H^\RVP44,#)z Returns ======= degree_m: int The degree_m is calculated as 1 + \sum_1 ^ n (d_i - 1), where d_i is the degree of the i polynomial c32"TF q^, xK R#5i)Nr_)rds& rr2MacaulayResultant._get_degree_m..Ls3l1uuls)sumrr1s&rrMacaulayResultant._get_degree_mCs33dll3333r4c \VPVP,^, VP^, 4#)z Returns ======= size: int The size of set T. Set T is the set of all possible monomials of the n variables for degree equal to the degree_m )rrr!r1s&rrMacaulayResultant.get_sizeNs+ .2DFFQJ??r4c \VPV4Uu.uF p\V!NK pp\VR\ RVP4R7#uupi)zO Returns ======= monomials: list A list of monomials of a certain degree. TrXrY)rrrrdr )r'degreemonomialrhs&& rr1MacaulayResultant.get_monomials_of_certain_degreeZs`6dnn6<>?>)1S(^> ?i&udnn=? ? ?sAc .p.p\VP4EFpV^8XdIVPVPV,, pVP V4pVP V4KSVP VP V^, ,VPV^, ,,4VPVPV,, pVP V4pVF7pVF.p\W4^8XgKVU u.uF p W8wgK V NK pp K0 K9 VP V4EK! V#uup i)zW Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix )r#r!rrrr:rr) r'row_coefficients divisiblerrr poss_rowsdivpitems & rget_row_coefficients&MacaulayResultant.get_row_coefficientsis tvvAAva8??G ''1  A!6!%a!e!4"56a8 @@H $C&q;!+:C)7)$,0I*.)I)7I'% !'' 2  )7s D? D? c .pVP4p\VP4FpW#,Fxp.p\VPV,V,.VP O5!pVP F#pVPVPV44K% VPV4Kz K \V4pV#)zL Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rr#r!rrrrr:rer) r'r>rr multiplier coefficientsrmonomacaulay_matrixs & r get_matrixMacaulayResultant.get_matrixs446tvvA.11 ! D,,Q/*<-!^^-!--D ''(;(;D(AB. L)2!,r4c  .pVPFkp.p\VP4F<wrEVP\ \ W%4VP V,844K> VPV4Km \V4UUu.uF*wrF\V4VP^, 8gK(VNK, ppp\V4UUu.uF*wrF\V4VP^, 8gK(VNK, pppWx3#uuppiuuppi)ak Returns ======= reduced: list A list of the reduced monomials non_reduced: list A list of the monomials that are not reduced Definition ========== A polynomial is said to be reduced in x_i, if its degree (the maximum degree of its monomials) in x_i is less than d_i. A polynomial that is reduced in all variables but one is said simply to be reduced. ) r enumeraterr:boolrrrr!) r'rr"r?rvrreduced non_reduceds & rget_reduced_nonreduced(MacaulayResultant.get_reduced_nonreduceds$ ""AD!$..1 Da!3t||A!FGH2   T " # "+9!5+!5!ftvvz)1!5+%.y%9/%9TQa&DFFAI-q%9 /## +/s %D6D %D 7D c NVP4wr#V.8Xd \^.4#\VP4UUu.uFwrEWPPV,,NK ppp\ VP 4Uu.uF+pVPV,PWd,4NK- ppVRV3,p.p \ VP4F8p VU u.uFqWR3,9NK p p RV 9gK'V PV 4K: WV3,#uuppiuupiuup i)z Returns ======= macaulay_submatrix: Matrix The Macaulay denominator matrix. Columns that are non reduced are kept. The row which contains one of the a_{i}s is dropped. a_{i}s are the coefficients of x_i ^ {d_i}. rbT) rrrrrr#r!rcoeffr>r:) r'rrrrrr reduction_setaisreduced_matrixrlrzaichecks && r get_submatrixMacaulayResultant.get_submatrixs$ $::< b=9 &dnn57537!ll1o--5 7dff '%1"(()9:% ' 7 +,,-Cq&11CE@5  C . K'((7' As$D<1DD")rrrrr!rrN)rrrrrr)rrrrrrrrrrs@rrrs:-\P0 4 @ ? 8.$>))r4rN)rmathrsympy.core.mulrsympy.matrices.denserrsympy.polys.polytoolsrrrsympy.simplify.simplifyr sympy.tensor.indexedr sympy.polys.monomialsr r sympy.polys.orderingsr rr(sympy.functions.combinatorial.factorialsr itertoolsrsympy.utilities.exceptionsrrrr_r4rrsL /::,,=.>=3@a4a4F])])r4