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Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation
- Source :
- Journal of Symbolic Computation, Journal of Symbolic Computation, 2020, 96, pp.85-107. ⟨10.1016/j.jsc.2019.02.013⟩, Journal of Symbolic Computation, Elsevier, 2020, 96, pp.85-107. ⟨10.1016/j.jsc.2019.02.013⟩
- Publication Year :
- 2020
- Publisher :
- HAL CCSD, 2020.
-
Abstract
- International audience; Sylvester doubles sums, introduced first by Sylvester (see Sylvester (1840, 1853)), are symmetric expressions of the roots of two polynomials. Sylvester’s definition of double sums makes no sense in the presence of multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aim of this paper is to give a new definition of Sylvester double sums making sense in the presence of multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove that double sums indexed by (a, b) are equal up to a constant if they share the same value for a+b, as well a proof of the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known (see Lascoux and Pragacz (2002); d’Andrea et al. (2007); Roy and Szpirglas (2011)). The more general proofs given here are using generalized Vandermonde determinants and symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of P and Q.
- Subjects :
- [MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC]
Root (chord)
MathematicsofComputing_NUMERICALANALYSIS
010103 numerical & computational mathematics
Mathematical proof
01 natural sciences
[ MATH.MATH-AC ] Mathematics [math]/Commutative Algebra [math.AC]
Combinatorics
13P15
Mathematics - Algebraic Geometry
Hermite interpolation
Simple (abstract algebra)
ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION
FOS: Mathematics
Computer Science::Symbolic Computation
0101 mathematics
Remainder
Algebraic Geometry (math.AG)
Mathematics
Sequence
Algebra and Number Theory
010102 general mathematics
subresultants
Sylvester double sums
generalized Vandermonde determinants
multivariateHermiteinterpolation
Vandermonde matrix
multivariate Hermite interpolation
Computational Mathematics
13P15, 13P05
Constant (mathematics)
Sylvesterdoublesums
Subjects
Details
- Language :
- English
- ISSN :
- 07477171 and 1095855X
- Database :
- OpenAIRE
- Journal :
- Journal of Symbolic Computation, Journal of Symbolic Computation, 2020, 96, pp.85-107. ⟨10.1016/j.jsc.2019.02.013⟩, Journal of Symbolic Computation, Elsevier, 2020, 96, pp.85-107. ⟨10.1016/j.jsc.2019.02.013⟩
- Accession number :
- edsair.doi.dedup.....91d115e84b8f6a3c642e921df1c18753
- Full Text :
- https://doi.org/10.1016/j.jsc.2019.02.013⟩