An upper bound on the number of zeros of a piecewise polinomial function
For mathematicians studying spline theory and related conjectures, this provides a theoretical result with a counterexample clarifying the conjecture's limits.
The paper establishes a precise relationship between the knots of a univariate spline and the number and distribution of its zeros. It proves a conjecture by De Concini and Procesi in the univariate, unimodular case, but shows it is false without unimodularity via a counterexample.
A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case. As a supplement, the same conjecture is shown, through computing a counterexample, to be false when unimodularity hypothesis is dropped.