Expressing a general form as a sum of determinants

• Autores: Luca Chiantini , Anthony V. Geramita
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 2, 2015, págs. 227-242
• Idioma: inglés
• DOI: 10.1007/s13348-014-0117-8
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• Resumen

  • Let A= (a_{ij}) be a non-negative integer k\times k matrix. A is a homogeneous matrix if a_{ij} + a_{kl}=a_{il} + a_{kj} for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in \mathbb {C}[x_1, \dots x_n] with deg(F) = \mathrm{trace}(A), what is the least integer, s(A), so that F = detM_1 + \cdots + detM_{s(A)}, where the M_i = (F^i_{lm}) are k\times k matrices of forms and deg F^i_{lm} = a_{lm} for every 1\le i \le s(A)? We consider this problem for n\ge 4 and we prove that s(A) \le k^{n-3} and s(A)