Recursive Polynomial Remainder Sequence and the Nested Subresultants (original) (raw)
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Subresultants in Recursive Polynomial Remainder Sequence
2008
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," and investigate their properties. In calculating PRS, if there exists the GCD (greatest common divisor) of initial polynomials, we calculate "recursively" with new PRS for the GCD and its derivative, until a constant is derived. We call such a PRS a recursive PRS. We define recursive subresultants to be determinants representing the coefficients in recursive PRS by coefficients of initial polynomials. Finally, we discuss usage of recursive subresultants in approximate algebraic computation, which motivates the present work.
Recursive polynomial remainder sequence and its subresultants
Journal of Algebra, 2008
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and "recursive subresultant," along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is built-up just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a "nested" expression, i.e. a Sylvester matrix whose elements are themselves determinants.
Matrix computation of subresultant polynomial remainder sequences in integral domains
Reliable Computing, 1995
We present an improved variant of the matrix-triangularization subresultant prs method [2] for the computation of a greatest common divisor of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder sequence. It is improved in the sense that we obtain complete theoretical results, independent of Van Vleck's theorem [13] (which is not always true [1], [6]), and, instead of transforming a matrix of order 2•max(m, n) [2], we are now transforming a matrix of order m + n. An example is also included to clarify the concepts.
2014
In 1971 using pseudo-divisions that is, by working in [x] Brown and Traub computed Euclid's polynomial remainder sequences (prs's) and (proper) subresultant prs's using sylvester1, the most widely known form of Sylvester's matrix, whose determinant defines the resultant of two polynomials. In this paper we use, for the first time in the literature, the Pell-Gordon Theorem of 1917, and sylvester2, a little known form of Sylvester's matrix of 1853 to initially compute Sturm sequences in [x] without pseudo-divisions that is, by working in É[x]. We then extend our work in É[x] and, despite the fact that the absolute value of the determinant of sylvester2 equals the absolute value of the resultant, we construct modified subresultant prs's, which may differ from the proper ones only in sign.
Subresultant Polynomial Remainder Sequences Obtained by Polynomial Divisions in Q[x] or in Z[x]
2016
In this paper we present two new methods for computing the subresultant polynomial remainder sequence (prs) of two polynomials f, g ∈ Z[x]. We are now able to also correctly compute the Euclidean and modified Euclidean prs of f, g by using either of the functions employed by our methods to compute the remainder polynomials. Another innovation is that we are able to obtain subresultant prs’s in Z[x] by employing the function rem(f, g, x) to compute the remainder polynomials in [x]. This is achieved by our method subresultants_amv_q (f, g, x), which is somewhat slow due to the inherent higher cost of com- putations in the field of rationals. To improve in speed, our second method, subresultants_amv(f, g, x), computes the remainder polynomials in the ring Z[x] by employing the function rem_z(f, g, x); the time complexity and performance of this method are very competitive. Our methods are two different implementations of Theorem 1 (Section 3), which establishes a one-to-one corresponde...
A sympy/sage Module for Computing Polynomial Remainder Sequences: [preprint]
2017
The Modified Euclidean prs is obtained by modifying the sign of the remainder of each polynomial division performed for the computation of the Euclidean prs. Analogously, the Modified Subresultant prs is obtained by modifying the matrix from which the Subresultant prs is obtained. Even though prs’s (c) and (d) are computed by evaluating sub-determinants of given matrices, our objective is to compute all four prs’s using the same type of polynomial divisions over the ring Z[x]. Our objective is not at all trivial and has eluded the efforts of great mathematicians, as our brief review below indicates. Initially, Collins, Brown and Traub [8], [9], [11], [12] used the so called prem pseudo-remainder function defined by LC(g) · f = q · g + h, (1)
Linear recurring sequences for computing the resultant of multivariate polynomials
Journal of Computational and Applied Mathematics, 1993
The effective computation of resultants of multivariate polynomials plays a major role in symbolic computation. Modular methods provide efficient approaches, of which Collins' algorithm is one of the best known. This paper presents an implementation of an algorithm based on linear recurring sequences for computing the resultant of two polynomials over a unique factorization domain. This method computes the resultant by means of a recursive function over a finite sequence with 2n-1 elements, where n is the maximum degree of the given polynomials. Computing-time analysis shows that the modular version of this approach and Collins' algorithm for computing resultants have the same complexity. The implementation of this algorithm is given in Maple.
Matrix computation of subresultant polynomial remainder
1995
We present an impr{wed variant of the matrix-triangularization subresultant prs method [1] fi~r the computation of a greatest comnum divi~w of two polynomials A and B (of degrees m and n, respectively) along with their polynomial remainder ~quence. It is impr~wed in the sense that we obtain complete theoretical results, independent {}f Van Vleck's theorem [13] (which is not always tnle [2, 6]), and, instead of transfornfing a matrix of order 2 .max(m, n) [1], we are now transforming a matrix of order m+ n. An example is al.,a~ induded to clarify the concepts. MaTp qHOe SbIq CAeH e cy6pe3yAt, TaHTHBIX IIOAHHOMIIaAbHbIX nocAeAOBaTeABHOCTefl OCTaTKOB B HHTeFpaABHblX 06AaCTXX A. F. AKP//rrAc, E. K. AKPHTAC, F. I/'l. MAAAmOHOK Flpe~lcraBJ'lerl yJlyqnleHH/:,ll~ Bapl.laHT MaTpl.lqHo-rpHaHlyJl.qpH3alIHOHHOlO cy6pe3yJlbTanTHOl'O MeToaa no-:lllHOMtlaJlbl-n~x ,u~c:xea<>marem, ti<m'refi <RTraTKOB ([I['IO) [1] a;xa ma,~nc.aemta Hal.16O/lbnlero ~gSntexo ae-:lnTedl~! 21ByX MHOrOttdleH(}B m n B (CTelleHelTI m ii n C{R)TBC~'I'(YrBeHHO) C O}IHOBpeMeHHI~M HaxoxK31eHHeM HX ~lOl']. YJlytlnleHne 3aKdiiottaeTcI, l B TOM, qTO IlOdlyqeHl~I 3aKOHt,IeHHIMe TeOpeTrtqecKne pe3yJlt, TaTIM, ae3aBr, lCrlMl:,le OX T~}pCMbl Baa B.aer.a [13] (KOTopaa tie Bcerlta cupaBeaJmma, CM [fi, 6]). KI.~Me XOn~, B.~ecxo npe~6pa2o~anu~ Maxpnm,1 nopsaKa 2-max(m, n) [1] renepb upeo6pa2yexc~ Maxpnua nop~taKa D2 + '/Z. I"Ipe, llCTaB.rleH qlIUleHtibllYl IlptiMep ~ldl~l IIdl211oCTpalll.n,I 3THX IIOJIo)KeHHI~I.
Theory of multiple polynomial remainder sequence
Publications of the Research Institute for Mathematical Sciences, 1984
Given a set of polynomials {P£, D (#), • • • > Po m) U)}j with coefficients in an integral domain I, we can generate a sequence of sets of remainders {P^(x) 9 ..., Pf w) U)}, x=l,2 v ..