Another approach to decide on real root existence for univariate Polynomials, and a multivariate extension for 3-SAT

TL;DR

This paper introduces a novel algorithm for determining the existence of real roots in univariate polynomials and extends this approach to a multivariate setting for 3-SAT problems, linking polynomial factorization to satisfiability.

Contribution

It presents six theorems on univariate polynomials and a new algorithm for root existence, along with a transformation connecting 3-SAT to multivariate polynomial positivity.

Findings

Algorithm correctly identifies polynomials with no positive real roots.

Transformation accurately encodes 3-SAT instances into positive-coefficient polynomials.

Theoretical foundation linking polynomial factorization to satisfiability.

Abstract

We present six Theorems on the univariate real Polynomial, using which we develop a new algorithm for deciding the existence of atleast one real root for univariate integer Polynomials. Our algorithm outputs that no positive real root exists, if and only if, the given Polynomial is a factor of a real Polynomial with positive coefficients. Next, we define a transformation that transforms any instance of 3-SAT into a multivariate real Polynomial with positive coefficients, if and only if, the instance is not satisfiable.