# Decision problem for a class of univariate Pfaffian functions

## TL;DR

This paper presents a symbolic decision procedure for univariate Pfaffian functions, extending to E-polynomials and multivariate cases, with algorithms for zero encoding and sign determination.

## Contribution

It introduces a new symbolic method for decision problems involving univariate Pfaffian functions, including algorithms for E-polynomials and multivariate extensions.

## Key findings

- A symbolic procedure with computable complexity based on Sturm sequences.
- An effective algorithm for E-polynomials avoiding oracles.
- A new notion of Thom encoding for zeros of E-polynomials.

## Abstract

We address the decision problem for sentences involving univariate functions constructed from a fixed Pfaffian function of order $1$. We present a new symbolic procedure solving this problem with a computable complexity based on the computation of suitable Sturm sequences. For a general Pfaffian function, we assume the existence of an oracle to determine the sign that a function of the class takes at a real algebraic number. For E-polynomials, we give an effective algorithm solving the problem without using oracles and apply it to solve a similar decision problem in the multivariate setting. Finally, we introduce a notion of Thom encoding for zeros of an E-polynomial and describe an algorithm for their computation.