Expected term bases for generic multivariate Hermite interpolation

多元Hermite方法用于计数多项式系统的实数和复数解

The multivariate Hermite method for counting real and complex solutions to polynomial systems

本文提出一种用于计数多项式系统实数和复数解的多元Hermite方法，并提供在Macaulay2中的实现。

Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation

Sylvester doubles sums, introduced first by Sylvester 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 $(k,\ell)$ are equal up to a constant if they share the same value for $k+\…

The Set of Unattainable points for the Rational Hermite Interpolation Problem

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional complete intersection varieties of odd codimension, the number of them being equal to the minimum between the degrees of the numerator and denominator of the problem. Each of these equidimensional varieties can be further decomposed as a union of as many rational (irreducible) varieties as input data points. We exhibit algorithms and equations defining all the…

Hermite interpolation by piecewise polynomial surfaces with polynomial area element

This paper is devoted to the construction of polynomial 2-surfaces which possess a polynomial area element. In particular we study these surfaces in the Euclidean space $\mathbb R^3$ (where they are equivalent to the PN surfaces) and in the Minkowski space $\mathbb R^{3,1}$ (where they provide the MOS surfaces). We show generally in real vector spaces of any dimension and any metric that the Gram determinant of a parametric set of subspaces is a perfect square if and only if the Gram determinant of its orthogonal complement is a perfect square. Consequently the polynomial surfaces of a given d…

Multivariate Polynomial Factorization by Interpolation Method

Factorization of polynomials arises in numerous areas in symbolic computation. It is an important capability in many symbolic and algebraic computation. There are two type of factorization of polynomials. One is convention polynomial factorization, and the other approximate polynomial factorization. Conventional factorization algorithms use symbolic methods to get exact factors of a polynomial while approximate factorization algorithms use numerical methods to get approximate factors of a polynomial. Symbolic computation often confront intermediate expression swell problem, which lower the eff…

多元拉格朗日插值及Cayley-Bacharach定理的一个应用

Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem For it

本文深入研究了多元拉格朗日插值，提出了充分交的概念，利用Bezout定理等证明了沿充分交代数流形的多项式空间维数，并基于Mysovskikh定理给出了适定节点集存在性的刻画条件，最后通过推广的Cayley-Bacharach定理解决了低维代数流形上的插值问题。