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Determinant boundaries and gaps in determinantal complexity — arXiv:2611.03554

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arXiv (en)

Original

https://arxiv.org/abs/2611.03554

License (as archived)

CC0-1.0

Archived

2026-11-03T18:59:16.954Z

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66fa04072afeee39b4edc00964a74c604ccda032be13a594f119b27338abe7cb

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Determinant boundaries and gaps in determinantal complexity — arXiv:2611.03554 — arXiv (en), https://arxiv.org/abs/2611.03554 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_dc2ca9ffba62 (Metraipolis archive dc_dc2ca9ffba62, sha256 66fa04072afeee39b4edc00964a74c604ccda032be13a594f119b27338abe7cb)

@misc{dcdc2ca9ffba62,
  title = {Determinant boundaries and gaps in determinantal complexity — arXiv:2611.03554},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.03554},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_dc2ca9ffba62; sha256 66fa04072afeee39b4edc00964a74c604ccda032be13a594f119b27338abe7cb},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.03554. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Landsberg, Manivel, and Ressayre constructed determinant-boundary components whose defining forms, for odd n >= 3, are limits of n x n determinants but cannot themselves be represented by determinants of that size. Their border determinantal complexity (DC) is therefore smaller than their exact DC. We prove that the difference can grow linearly with n. For every n >= 10, we construct an explicit homogeneous form of degree n, concise in n^2 variables, whose orbit closure is a boundary component, whose border DC is n, and whose exact DC is at least 2n - 1. Thus n rows suffice when limits are allowed, whereas every exact representation requires at least 2n - 1 rows. The same bound holds for the unpadded odd-skew forms at every odd n >= 3. We also prove that every concise degree-n boundary form in n^2 variables has exact DC at least n + 2. We construct three families of pairwise distinct boundary components. Together with the traceless determinant, they give at least floor((n-1)/2) + floor(n/3) components for every n >= 3, with an additional term counted explicitly in the paper. The main lower bound follows from a general degree lemma. If an affine matrix represents an irreducible homogeneous polynomial f of degree d and has globally free rank-one cokernel over the coordinate ring of f = 0, its size is at least 2d - 1. The adjugate factors into two nonzero kernel vectors, each requiring a coordinate of degree at least d - 1. Comparing their product with the degree of a cofactor proves the bound. The lemma is sharp for individual representations, as witnessed by algebraic branching programs. Whether the exact-versus-border gap can grow superlinearly remains open.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv cs.CC, nuovi lavori del giorno).
Categorie arXiv: Computational Complexity (cs.CC).
Pagina del lavoro: https://arxiv.org/abs/2611.03554 — 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 2026-11-03T18:18:37.682Z dalla pagina https://arxiv.org/list/cs.CC/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).