Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities
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 ($\leq 0$); 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 $\{x\in \mathbb{R} ^n\mid x^T Ax \mathbin{\star } 0\}\cap \{x\in \mathbb{R} ^n\mid x^T Bx \mathbin{\#} 0\}\subset \{0\}$, where $\star $ and $\#$ can be any of $\{<,\leq ,=\}$, 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) \mathbin{\star } 0\}\cap \{x\in \mathbb{R} ^n\mid g(x) \mathbin{\#} 0\}=\emptyset $ 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\}=\emptyset $; and the nonhomogeneous (strict) Finsler lemma to determine whether $\{f(x)\leq 0\}\cap \{g(x)=0\}=\emptyset .$ The paper provides the answers to both, in theorems and algorithms.