THE ROTA METHOD FOR SOLVING POLYNOMIAL EQUATIONS: A MODERN APPLICATION OF INVARIANT THEORY

Abstract

Algebraic invariants are defined for the purpose of gaining insights into solving polynomial equations. Polynomial invariants are disclosed here as an alternative to and to clarify the umbral method of Gian-Carlo Rota. A process for solving cubic polynomial equations is examined and extended to quintic (or 5th degree) polynomial equations. It is proved that a general cubic polynomial is “apolar” to a quadratic polynomial. It is proved that a quadratic polynomial and cubic polynomial which are apolar either both have repeated roots or both have distinct roots. In the case of repeated roots, these roots are shared by the cubic and quadratic polynomials that are apolar. In the case, in which the derived quadratic which is apolar to a given cubic has distinct roots, it is shown the cubic polynomial px may be transformed to px = c1x − r1 + c2x − r2, where r1 and r2 are the distinct roots of the quadratic polynomial. This will allow the roots of the cubic to be found using algebraic operations. It will be shown that these methods can be extended to show that a given quintic polynomial is in general apolar to a cubic polynomial. Some remaining questions are posed at the end of the article. AMS Subject Classification: 13A50