Homology of spaces of non-resultant polynomial systems in R^2 and C^2

TL;DR

This paper computes the cohomology groups of spaces of polynomial systems in R^2 and C^2 that do not have non-trivial solutions, providing algebraic topological insights into these spaces.

Contribution

It determines the integer and rational cohomology groups of spaces of non-resultant polynomial systems in R^2 and C^2, extending understanding of their topological structure.

Findings

Computed integer cohomology groups for R^2 systems

Calculated rational cohomology groups for C^2 systems

Extended algebraic topology knowledge of polynomial system spaces

Abstract

The resultant veriety in the space of systems of homogeneous polynomials of given degrees consists of such systems having non-trivial solutions. We calculate the integer cohomology groups of all spaces of non-resultant systems of polynomials $R^2 \to R$, and also the rational cohomology groups of similar systems in $C^2$.