# Which quartic polynomials have a hyperbolic antiderivative?

## TL;DR

This paper characterizes which quartic polynomials with real zeros are derivatives of polynomials with all real zeros, providing a precise quadratic inequality condition involving their zeros.

## Contribution

It establishes a necessary and sufficient quadratic form inequality condition for quartic polynomials to have hyperbolic antiderivatives.

## Key findings

- Derived a quadratic inequality condition for quartic polynomials
- Identified when a quartic polynomial is a derivative of a hyperbolic polynomial
- Clarified the limitations of derivatives of polynomials with all real zeros

## Abstract

Every linear, quadratic or cubic polynomial having all real zeros is the derivative of a polynomial having all real zeros. The statement is false for higher degree polynomials. In particular, not every fourth degree polynomial with real zeros is the derivative of a polynomial having all real zeros. We derive a necessary and sufficient condition for a quartic polynomial to be the derivative of a polynomial having all real zeros. This condition is a single quadratic form inequality involving the zeros of the quartic polynomial.