Positivity of continuous piecewise polynomials (Q2903276)

Consider a simplicial complex \(\Delta= \sigma_1 \cup \cdots \cup\sigma_k\) in \({\mathbb R}^n\), and let \(C^0(\Delta)\) denote the algebra of all continuous piecewise polynomials on \(\Delta\), consisting of all real continuous functions \(F\) on \(\Delta\) whose restriction to each \(\sigma_j\) is a polynomial. The article under review gives Positivstellensätze (i.e., certificates of positivity) for functions \(F\in C^0(\Delta)\) positive on \(\Delta\). To do this, the author uses tent functions describing \(C^0(\Delta)\). Here are two main results of this paper. The first one is a Putinar-type Positivstellensatz [\textit{M. Putinar}, Indiana Univ. Math. J. 42, No. 3, 969--984 (1993; Zbl 0796.12002)] for functions \(F\in C^0(\Delta)\) \textit{strictly} positive on \(\Delta\), cf.~Theorem 2.2. The second main result (Theorem 2.6) is a strengthening to one-dimensional simplicial complexes, where the author obtains a Nichtnegativstellensatz with degree bounds.