Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities

Abstract

Given two quadratic functions \( f(x) = x^T Ax + 2a^T x + a_0 \) and \( g(x) = x^T Bx + 2b^T x + b_0 ,\) each associated with either the strict inequality ($<0$); non-strict inequality ($\le 0$); or the equality ($=0$), it is a fundamental question to ask whether or not the joint system has a solution. For homogeneous quadratic systems ($a=b=0,a_0=b_0=0$), starting from Finsler's lemma in 1936 until Yuan's alternative lemma in 1990, all combinations of the unsolvability for $\{x\in {\mathbb{R}}^n\mid x^{T}Ax\star 0\}\cap \{x\in {\mathbb{R}}^n\mid x^{T}Bx\#0\}\subset \{0\}$, where $\star$ and $\#$ can be any of $\{<,\le ,=\}$, have been shown to possess either a positive definite or a positive semi-definite matrix pencil of $A$ and $B.$ Extensions to nonhomogeneous quadratic systems $\{x\in {\mathbb{R}}^n\mid f(x)\star 0\}\cap \{x\in {\mathbb{R}}^n\mid g(x)\#0\}=\varnothing$ have been done for several cases already. Two challenging cases remain open: the nonhomogeneous Calabi Theorem which determines when $\{f(x)=0\}\cap \{g(x)=0\}=\varnothing$; and the nonhomogeneous (strict) Finsler lemma to determine whether $\{f(x)\le 0\}\cap \{g(x)=0\}=\varnothing .$ The paper provides the answers to both, in theorems and algorithms.