When are the roots of a polynomial real and distinct? A graphical view

TL;DR

This paper provides a visual and intuitive proof of the classical criterion for real and distinct polynomial roots, explores its generalization to entire functions, and discusses implications for the Riemann Hypothesis.

Contribution

It offers a graphical perspective on the classical roots criterion and extends the discussion to entire functions, challenging existing interpretations of the Riemann Hypothesis.

Findings

Polynomial roots are real and distinct iff the polynomial and all derivatives have only negative minima and positive maxima.

Graphical intuition aids understanding of classical root criteria.

Generalization to entire functions suggests a different view on the Riemann Hypothesis.

Abstract

We prove the classical result, which goes back at least to Fourier, that a polynomial with real coefficients has all zeros real and distinct if and only if the polynomial and also all of its nonconstant derivatives have only negative minima and positive maxima. Intuition for the result, involving illuminating pictures, is described in detail. The generalization of Fourier's theorem to certain entire functions of order one (which is conjectural) suggests that the official description of the Riemann Hypothesis Millennium Problem incorrectly describes an equivalence to the Riemann Hypothesis. The paper is reasonably self-contained and is intended be accessible (possibly with some help) to students who have taken two semesters of calculus.