On Waring's Problem for Systems of Skew-Symmetric Forms.

摘要

In 1707, E. Waring suggested the problem of expressing every positive integer as a sum of at most s(d) d-th powers of positive integers. This problem was affirmatively solved by Hilbert in 1909.;In this dissertation we discuss a similar question for systems of skew-symmetric forms which asks: "What is the smallest integer s( m, n, k) such that the generic (m + 1) skew-symmetric degree k + 1 forms in (n + 1) variables defined over an algebraic closed field are expressible as linear combinations of the same s(m, n, k) (k + 1)-st powers of linear forms?" This problem is known as Waring's problem for systems of skew-symmetric forms..;It is known that s(m, n, k) should be at least (m + 1) &parl0;n+1k+1&parr0; / (m + (k + 1) (n -- k) + 1). The two main goals of this dissertation are to show the existence of triples (m, n, k) such that s( m, n, k) is strictly bigger than the above-mentioned integer and to establish for some families of triples (m, n, 1) that s(m, n, 1) is actually equal to that integer.;Waring's problem for systems of skew-symmetric forms can be naturally translated into a classical problem in algebraic geometry. In this dissertation, we will describe how algebraic varieties can be associated to the collection of all systems of skew-symmetric forms with a given degree, number of equations and number of variables. We will then use algebro-geometric approaches to establish the existence of cases where s(m, n, k ) does not have the expected value.