An upper bound on the number of zeros of a piecewise polinomial function

TL;DR

This paper establishes an upper bound on the zeros of univariate splines based on their knots, proving a conjecture in a special case and providing a counterexample when conditions are relaxed.

Contribution

It introduces a precise relationship between spline knots and zeros, confirming a conjecture in the unimodular case and disproving it otherwise.

Findings

Confirmed the conjecture for unimodular univariate splines.

Disproved the conjecture in the non-unimodular case with a counterexample.

Established a bound linking knot positions to zero distribution.

Abstract

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.