Zero counting for a class of univariate Pfaffian functions

TL;DR

This paper introduces a new method for counting real zeros of certain univariate Pfaffian functions using Sturm sequences, with an efficient algorithm for E-polynomials that avoids oracles and provides bounds on zeros.

Contribution

It develops a novel Sturm sequence-based procedure for univariate Pfaffian functions and an oracle-free algorithm for E-polynomials with exponential complexity bounds.

Findings

New Sturm sequence construction for Pfaffian functions

Oracle-free algorithm for E-polynomials

Explicit bounds on zeros of E-polynomials

Abstract

We present a new procedure to count the number of real zeros of a class of univariate Pfaffian functions of order $1$. The procedure is based on the construction of Sturm sequences for these functions and relies on an oracle for sign determination. In the particular case of $E$-polynomials, we design an oracle-free effective algorithm solving this task within exponential complexity. In addition, we give an explicit upper bound for the absolute value of the real zeros of an $E$-polynomial.