Extension of the two-variable Pierce-Birkhoff conjecture to generalized polynomials - Mathematics > Algebraic GeometryReportar como inadecuado




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Abstract: Let R denote the reals, and let h: R^n -> R be a continuous,piecewise-polynomial function. The Pierce-Birkhoff conjecture 1956 is thatany such h is representable in the form sup i inf j f {ij}, for some finitecollection of polynomials f {ij} in Rx 1,

.,x n. A simple example is hx 1= |x 1| = sup{x 1 -x 1}. In 1984, L. Mahe and, independently, G. Efroymson,proved this for n < 3; it remains open for n > 2. In this paper we prove ananalogous result for -generalized polynomials- also known as signomials,i.e., where the exponents are allowed to be arbitrary real numbers, and notjust natural numbers; in this version, we restrict to the positive orthant,where each x i > 0. As before, our methods work only for n < 3.



Autor: Charles N. Delzell

Fuente: https://arxiv.org/







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