15,229 research outputs found
On the representation of integers by quadratic forms
Let Q be a non-singular quadratic form with integer coefficients. When Q is
indefinite we provide new upper bounds for the least non-trivial integral
solution to the equation Q=0. When Q is positive definite we provide improved
upper bounds for the least positive integer k such that the equation Q=k is
insoluble in integers, despite being soluble modulo every prime power.Comment: 33 page
Isoparametric polynomials and sums of squares
The Hilbert's 17th problem asks that whether every nonnegative polynomial can
be a sum of squares of rational functions. It has been answered affirmatively
by Artin. However, as to the question whether a given nonnegative polynomial is
a sum of squares of polynomials is still a central question in real algebraic
geometry. In this paper, we solve this question completely for the nonnegative
polynomials associated with isoparametric polynomials (initiated by E. Cartan)
which define the focal submanifolds of the corresponding isoparametric
hypersurfaces.Comment: 38 page
Inertia properties of indefinite quadratic forms
We study the relation between the solutions of two estimation problems with indefinite quadratic forms. We show that a complete link between both solutions can be established by invoking a fundamental set of inertia conditions. While these inertia conditions are automatically satisfied in a standard Hilbert space setting, they nevertheless turn out to mark the differences between the two estimation problems in indefinite metric spaces. They also include, as special cases, the well-known conditions for the existence of H-infinity-filters and controlers
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