Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities
Abstract
Given two quadratic functions f(x) = xT Ax + 2aT x + a0 and g(x) = xT Bx + 2bT x + b0 , each associated with either the strict inequality (<0); non-strict inequality (≤ 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, a0=b0=0), starting from Finsler's lemma in 1936 until Yuan's alternative lemma in 1990, all combinations of the unsolvability for {x∈ ℝ n| xT Ax mathbin⋆ 0}∩ {x∈ ℝ n| xT Bx mathbin# 0}⊂ {0}, where $\star $ and # can be any of {<,≤ ,=}, 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)≤ 0}∩ {g(x)=0}=∅ . The paper provides the answers to both, in theorems and algorithms.
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BibTeX
@misc{wang2026last,
title = {{Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities}},
author = {Min-Chi Wang and Ruey-Lin Sheu and Huu-Quang Nguyen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30571},
eprint = {2608.30571},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30571},
}
RIS
TY - UNPB
TI - Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities
AU - Min-Chi Wang
AU - Ruey-Lin Sheu
AU - Huu-Quang Nguyen
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30571
UR - https://arxiv.org/abs/2608.30571
AB - 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 (≤ 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∈ ℝ ^n| x^T Ax mathbin⋆ 0}∩ {x∈ ℝ ^n| x^T Bx mathbin# 0}⊂ {0}, where $\star $ and # can be any of {<,≤ ,=}, 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)≤ 0}∩ {g(x)=0}=∅ . The paper provides the answers to both, in theorems and algorithms.
ER -