Positive bases in spaces of polynomials

TL;DR

This paper investigates the maximum dimension of polynomial subspaces with positive bases on a compact set D, revealing how topological features of D influence this maximum and showing that full dimension is rarely achievable.

Contribution

It determines the maximal dimension of polynomial subspaces with positive bases depending on the topology of D, providing exact values and conditions for various cases.

Findings

Maximum dimension depends on topological features of D

Full dimension m+1 is only possible for low m or finite D

No set D allows positive bases for all polynomial degrees m

Abstract

For a nonempty compact set D of R we determine the maximal possible dimension of a subspace X of polynomial functions of degree at most m which possesses a positive bases (where positivity is understood on D). The exact value of this maximal dimension depends on topological features of the base set D. We show that at many cases dimension m can be achieved.Whereas only for low m or finite sets D it is possible to have m+1 (the full dimension of the space of polynomials of degree at most m) dimensional subspace X with positive basis. Therefore, it turns out that for no D it is possible to have a positive basis of the polynomial space of degree at most m for all m in N.