New and Old Results in Resultant Theory

TL;DR

This paper surveys the landscape of resultant and discriminant theory, emphasizing explicit, computable formulas across multiple formalisms (Sylvester, Bezout, Macaulay, and Schur) and extending to invariant- and polytope-based perspectives. It connects purely algebraic constructs to physics-relevant non-Gaussian integrals through Ward identities and integral discriminants, highlighting action-independence and hypergeometric structures. The work highlights practical computation pathways for non-linear systems, including determinant- and ratio-based representations, as well as geometric interpretations via Newton polytopes and volume calculations, while outlining open directions such as Macaulay matrices and GKZ methods. Overall, it positions resultant theory as a versatile toolkit for non-linear algebra with potential applications in theoretical physics and beyond.

Abstract

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.

Figures (9)

• Figure 1: Determinant of matrix $S_{ij}$, represented as a diagram of tensor contraction. Black 2-valent vertices represent tensor $S$, white n-valent vertices represent tensor $\epsilon$.
• Figure 2: The degree 4 invariant $I_4$ of a 3-form in 2 variables, represented as a diagram of tensor contraction. Black 3-valent vertices represent tensor $S$, white 2-valent vertices represent tensor $\epsilon$.
• Figure 3: The degree 2 invariant $I_2$ of a 4-form in 2 variables, represented as a diagram of tensor contraction. Black 4-valent vertices represent tensor $S$, white 2-valent vertices represent tensor $\epsilon$.
• Figure 4: The degree 3 invariant $I_3$ of a 4-form in 2 variables, represented as a diagram of tensor contraction. Black 4-valent vertices represent tensor $S$, white 2-valent vertices represent tensor $\epsilon$.
• Figure 5: The degree 4 invariant $I_4$ of a 5-form in 2 variables, represented as a diagram of tensor contraction. Black 5-valent vertices represent tensor $S$, white 2-valent vertices represent tensor $\epsilon$.