\h;SrSSKJr SSKJr SSKJrJr SSKJ r J r J r SSK J r SSKJr SSKJrJr SS KJr SS KJrJr SS KJr SS KJr SS KJr "SS5r"SS5rg)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_warningc^\rSrSrSrSr\S5rSrSr Sr Sr S r S r S rS rS rg)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^XlX l[UR5Ul[UR5Ul[ S5n[ UR5Vs/sHoCUPM snUl[ UR5V^s/sH"m[U4SjUR55PM$ snUl gs snfs snf)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 alphac3@># UHn[U5Tv M g7f)N)r).0polyis [/var/www/auris/envauris/lib/python3.13/site-packages/sympy/polys/multivariate_resultants.py *DixonResultant.__init__..[s SBR$T!21!5BRsN) polynomials variableslennmr rangedummy_variablesmax _max_degrees)selfr r!ars `r__init__DixonResultant.__init__Ds'"T^^$T%%&  .3DFFms (A A%c TURU5n[URU5n[US[ SUR5S9n[ UR 5VVs/sH9nUVs/sH(n[U/URQ76RU5PM* snPM; snn5nURSURS:waQ[URS5Vs/sH%n[SUSS2U455(dM#UPM' nnUSS2U4nU$s snfs snnfs snf) z Construct the Dixon matrix from the coefficients of polynomial \alpha. Each coefficient is viewed as a polynomial of x_1, ..., x_n. Tlexreversekeyrr8c3*# UH oS:gv M g7frN)relements rr2DixonResultant.get_dixon_matrix..s42-4a<2N) rXr r!sortedrrrTrcoeff_monomialshaper%any) r)rUr4 monomialscr$ dixon_matrixcolumnkeeps rget_dixon_matrixDixonResultant.get_dixon_matrixs4**:6 "$..+> 9d+E4>>BD )3):):)<>)?   a L$6$6q$9 9).|/A/A"/E)F5)Fv4'6 244)FD5(40L 4> 5s$ D )/DD !"D%D%D c j^^TR(agTRup#[TR5S5m[ U5V^s/sH+m[ UU4Sj[ U555(dM)TPM- nnTUSS24m[ S/US- -S/-/5nTSSS24U:Xaggs snf)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]. Frc38># UHnTTU4S:gv M g7frarb)rjrmatrixs rr2DixonResultant.KSY_precondition..s*Oh6!Q$<1+>> from sympy import symbols >>> from sympy.polys.multivariate_resultants import MacaulayResultant >>> x, y, z = symbols('x, y, z') >>> a_0, a_1, a_2 = symbols('a_0, a_1, a_2') >>> b_0, b_1, b_2 = symbols('b_0, b_1, b_2') >>> c_0, c_1, c_2,c_3, c_4 = symbols('c_0, c_1, c_2, c_3, c_4') >>> f = a_0 * y - a_1 * x + a_2 * z >>> g = b_1 * x ** 2 + b_0 * y ** 2 - b_2 * z ** 2 >>> h = c_0 * y * z ** 2 - c_1 * x ** 3 + c_2 * x ** 2 * z - c_3 * x * z ** 2 + c_4 * z ** 3 >>> mac = MacaulayResultant(polynomials=[f, g, h], variables=[x, y, z]) >>> mac.monomial_set [x**4, x**3*y, x**3*z, x**2*y**2, x**2*y*z, x**2*z**2, x*y**3, x*y**2*z, x*y*z**2, x*z**3, y**4, y**3*z, y**2*z**2, y*z**3, z**4] >>> matrix = mac.get_matrix() >>> submatrix = mac.get_submatrix(matrix) >>> submatrix Matrix([ [-a_1, a_0, a_2, 0], [ 0, -a_1, 0, 0], [ 0, 0, -a_1, 0], [ 0, 0, 0, -a_1]]) See Also ======== Notebook in examples: sympy/example/notebooks. References ========== .. [1] [Bruce97]_ .. [2] [Stiller96]_ cFXlX l[U5UlURVs/sHn[ U/URQ76PM snUlUR 5UlUR5Ul URUR5Ul gs snf)z Parameters ========== variables: list A list of all n variables polynomials : list of SymPy polynomials A list of m n-degree polynomials N) r r!r"r#rdegrees _get_degree_mdegree_mget_sizemonomials_sizeget_monomials_of_certain_degree monomial_set)r)r r!rs rr+MacaulayResultant.__init__+s'"Y ++-+AE T;DNN;+- **, "mmo!@@O-s Bc@S[SUR55-$)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 r8c3*# UH oS- v M g7f)r8Nrb)rds rr2MacaulayResultant._get_degree_m..Ls3l1ulre)sumrr3s rrMacaulayResultant._get_degree_mCs33dll3333r6ch[URUR-S- URS- 5$)z Returns ======= size: int The size of set T. Set T is the set of all possible monomials of the n variables for degree equal to the degree_m r8)rrr#r3s rrMacaulayResultant.get_sizeNs+ .2DFFQJ??r6c[URU5Vs/sH n[U6PM nn[US[ SUR5S9$s snf)zO Returns ======= monomials: list A list of monomials of a certain degree. Tr[r\)rr!rrfr)r)degreemonomialrjs rr1MacaulayResultant.get_monomials_of_certain_degreeZs`6dnn6<>?>)1S(^> ?i&udnn=? ? ?sA c8/n/n[UR5HnUS:Xa@URURU- nUR U5nUR U5 MIUR UR US- URUS- -5 URURU- nUR U5nUH6nUH-n[X5S:XdMUV s/sH n X:wdM U PM nn M/ M8 UR U5 M U$s sn f)zW Returns ======= row_coefficients: list The row coefficients of Macaulay's matrix rr8)r%r#rrrr=r!r ) 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 0D cf/nUR5n[UR5HznX#Hon/n[URUU-/UR Q76nUR H#nURURU55 M% URU5 Mq M| [U5nU$)zL Returns ======= macaulay_matrix: Matrix The Macaulay numerator matrix ) rr%r#rr r!rr=rgr) r)rArr multiplier coefficientsrmonomacaulay_matrixs r get_matrixMacaulayResultant.get_matrixs446tvvA.1 ! D,,Q/*<-!^^-!--D ''(;(;D(AB. L)2!,r6c /nURHgn/n[UR5H8upEUR[ [ X%5UR U:55 M: URU5 Mi [U5VVs/sH%upF[U5URS- :dM#UPM' nnn[U5VVs/sH%upF[U5URS- :dM#UPM' nnnXx4$s snnfs snnf)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. r8) r enumerater!r=boolrrrr#) r)rr$rBrvrreduced non_reduceds rget_reduced_nonreduced(MacaulayResultant.get_reduced_nonreduceds$ ""AD!$..1 Da!3t||A!FGH2   T " # "+9!5+!5!ftvvz)!5+%.y%9/%9TQa&DFFAI-%9 /## +/s"C7.C7"C=+C=c  UR5up#U/:Xa [S/5$[UR5VVs/sHupEXPRU-PM nnn[ UR 5Vs/sH#nURURXd5PM% nnUSS2U4n/n [ UR5H6n UV s/sH oXSS24;PM n n SU ;dM%U RU 5 M8 XU4$s snnfs snfs sn f)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}. r8NT) rrrr!rr%r#r coeffrAr=) r)rtrrrr reduction_setaisreduced_matrixrnr|aichecks r get_submatrixMacaulayResultant.get_submatrixs! $::< b=9 &dnn57537!ll1o-5 7dff '%1"(()9:% ' 7 +,,-Cq&11CE@5  C . K'((7' AsD4*DD )rrrrr#r r!N)rrrrrr+rrrrrrrrrbr6rrrs2-\P0 4 @ ? 8.$>)r6rN)rmathrsympy.core.mulrsympy.matrices.denserrsympy.polys.polytoolsrrr sympy.simplify.simplifyr sympy.tensor.indexedr sympy.polys.monomialsr r sympy.polys.orderingsrrr(sympy.functions.combinatorial.factorialsr itertoolsrsympy.utilities.exceptionsrrrrbr6rrsL /::,,=.>=3@a4a4F])])r6