Symmetric interpolation, Exchange Lemma and Sylvester sums

ABSTRACT

The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bézout coefficients.

KEYWORDS:

• Subresultants
• Sylvester double sums
• symmetric Lagrange interpolation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13P15
• 65D05

Acknowledgments

Agnes Szanto and Teresa Krick thank the Simons Institute for the Theory of Computing, for the Fall’14 program “Algorithms and Complexity in Algebraic Geometry” where this work was started. We are also grateful to Carlos D’Andrea for the many useful discussions we had with him, to Giorgio Ottaviani for a great conversation on symmetric polynomials which improved the proof of Lemma 2.2, and to Ricky Ini Liu who helped us understanding the connections between Schur polynomials and Proposition 3.12.