Sturm Sequences and Modified Subresultant Polynomial Remainder Sequences

ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

ACM Computing Classification System (1998): G.1.1, G.1.2.

Three New Methods for Computing Subresultant Polynomial Remainder Sequences (PRS’S)

Given the polynomials f, g ∈ Z[x] of degrees n, m, respectively, with n > m, three new, and easy to understand methods — along with the more efficient variants of the last two of them — are presented for the computation of their subresultant polynomial remainder sequence (prs). All three methods evaluate a single determinant (subresultant) of an appropriate sub-matrix of sylvester1, Sylvester’s widely known and used matrix of 1840 of dimension (m + n) × (m + n), in order to compute the correct sign of each polynomial in the sequence and — except for the second method — to force its coefficients to become subresultants. Of interest is the fact that only the first method uses pseudo remainders. The second method uses regular remainders and performs operations in Q[x], whereas the third one triangularizes sylvester2, Sylvester’s little known and hardly ever used matrix of 1853 of dimension 2n × 2n. All methods mentioned in this paper (along with their supporting functions) have been implemented in Sympy and can be downloaded from the link http://inf-server.inf.uth.gr/~akritas/publications/subresultants.py

On some Extremal Problems of Landau

2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.

Complex Analogues of the Rolle's Theorem

2000 Mathematics Subject Classification: 30C10.

A Unified Group-Theoretic Method on Improper Partial Semi-Bilateral Generating Functions

2000 Mathematics Subject Classification: 33A65, 33C20.

The Lower Estimate for Bernstein Operator

MSC 2010: 41A10, 41A15, 41A25, 41A36

Extention of Apolarity and Grace Theorem

MSC 2010: 30C10

Hermite Series with Polar Singularities

MSC 2010: 33C45, 40G05

Class Number Two for Real Quadratic Fields of Richaud-Degert Type

2000 Mathematics Subject Classification: Primary: 11D09, 11A55, 11C08, 11R11, 11R29; Secondary: 11R65, 11S40; 11R09.

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45

On a Generalization of a Theorem due to Larcombe on the Sum of a 3F2 Series

MSC 2010: 33C20

On Some Generalizations of Classical Integral Transforms

MSC 2010: 44A15, 44A20, 33C60

New Coefficient Conditions for Functions Starlike with Respect to Symmetric Points

2000 Mathematics Subject Classification: Primary 30C45, secondary 30C80.

The Direct and Inverse Spectral Problems for some Banded Matrices

2000 Mathematics Subject Classification: 15A29.