Round Quadratic Forms.
Representations of non-negative polynomials having finitely many zeros
Consider a compact subset of real -space defined by polynomial inequalities . For a polynomial non-negative on , natural sufficient conditions are given (in terms of first and second derivatives at the zeros of in ) for to have a presentation of the form , a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in [11] as well as arguments from quadratic form theory...
Extending piecewise polynomial functions in two variables
We study the extensibility of piecewise polynomial functions defined on closed subsets of to all of . The compact subsets of on which every piecewise polynomial function is extensible to can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise polynomial...
Quotients of index two and general quotients in a space of orderings
We investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field ℚ(x) is particularly interesting, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index 2 generalizes...
Separating families for semi-algebraic sets.
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