连续分段多项式的正性
Positivity of continuous piecewise polynomials
摘要：
实代数几何通过将多项式表示为平方和和定义不等式的适当组合，为半代数集上的多项式的正性提供了证明。我们展示了如何Putinar的严格正多项式紧集定理可以应用在严格正的分段多项式的情况下，单纯复形。在1维情形下，我们改进了这个结果，使其覆盖所有非负的分段多项式，并给出了显式的度界。
英文摘要：
Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for strictly positive polynomials on compact sets can be applied in the case of strictly positive piecewise polynomials on a simplicial complex. In the 1-dimensional case, we improve this result to cover all non-negative piecewise polynomials and give explicit degree bounds.
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