Nullstellensätze and Applications

Abstract

In this expository paper, we present simple proofs of the Classical, Real, Projective and Combinatorial Nullstellensätze. Several applications are also presented such as a classical theorem of Stickelberger for solutions of polynomial equations in terms of eigenvalues of commuting operators, construction of a principal ideal domain which is not Euclidean, Hilbert's 17 th problem, the Borsuk-Ulam theorem in topology and solutions of the conjectures of Dyson, Erdös and Heilbronn. * This article grew out of discussions with late Prof. Dr. Uwe Storch (1940Storch ( -2017 and lectures delivered by the second and the third author in various workshops and conferences. Prof. Uwe Storch was known for his work in commutative algebra, analytic and algebraic geometry, in particular derivations, divisor class group and resultants.

Stickelberger and the Eigenvalue Theorem

“…When mathematicans say "Stickelberger's Theorem", they are usually referring to Theorem 2.1. This is probably what led the authors of [11] to cite [29] as the source for their version of the Eigenvalue Theorem.…”

“…The name "Stickelberger's Theorem" in Theorem 1.2 is from [12]. Versions of Theorems 1.1 and 1.2 also named "Stickelberger's Theorem" can be found in the papers [11,26,32], and [23] has a "Stickelberger's Theorem" for positive-dimensional solution sets. A "Stickelberger's Theorem" that focuses on (1.2) and (1.3) can be found in [2].…”

“…A common feature of these papers is that no reference to Stickelberger is given! An exception is [11], which refers to the wrong paper of Stickelberger. There is an actual theorem of Ludwig Stickelberger lurking in the background, in the paper Über eine neue Eigenschaft der Diskriminanten algebraischer Zahlkörper [28] that appeared in the proceedings of the first International Congress of Mathematicians, held in Zürich in 1897. This paper includes Theorems I-XIII, most dealing with traces and properties of the discriminant of a number field.…”