# IJPAM: Volume 89, No. 2 (2013)

**THE ROTA METHOD FOR SOLVING POLYNOMIAL**

EQUATIONS: A MODERN APPLICATION

OF INVARIANT THEORY

EQUATIONS: A MODERN APPLICATION

OF INVARIANT THEORY

Sul Ross State University

Alpine, TX 79830, USA

**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 may be transformed to where and 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.

**Received: **July 17, 2013

**AMS Subject Classification: **13A50

**Key Words and Phrases: **invariant theory, invariants, polynomial equations

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**DOI: 10.12732/ijpam.v89i2.4**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2013

**Volume:**89

**Issue:**2