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Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities — arXiv:2608.30571

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.30571
License (as archived): CC0-1.0
Archived: 2026-10-17T09:06:51.213Z
SHA-256 of the archived text: 0a4c1bbc4cd89bd53a40b343e9ba1f6c04bb8382cbd81314e5f628c28b288ffe
Stable URL of this page: https://metraipolis.com/documento/dc_da360bb7d3d7

How to cite

Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities — arXiv:2608.30571 — arXiv (en), https://arxiv.org/abs/2608.30571 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_da360bb7d3d7 (Metraipolis archive dc_da360bb7d3d7, sha256 0a4c1bbc4cd89bd53a40b343e9ba1f6c04bb8382cbd81314e5f628c28b288ffe)

BibTeX

@misc{dcda360bb7d3d7,
  title = {Last two pieces of the puzzle for unsolvability of a system of two quadratic (in)equalities — arXiv:2608.30571},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.30571},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_da360bb7d3d7; sha256 0a4c1bbc4cd89bd53a40b343e9ba1f6c04bb8382cbd81314e5f628c28b288ffe},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.30571. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-01 in math.OC).
Categorie arXiv: Optimization and Control (math.OC).
Pagina del lavoro: https://arxiv.org/abs/2608.30571 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T08:40:19.764Z dalla pagina https://arxiv.org/catchup/math.OC/2026-09-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).