Simon conjecture and the \text{ v }-number of monomial ideals

• Ficarra, Antonino ^[1]
  1. [1] University of Messina

     University of Messina

     Mesina, Italia

     
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 76, Fasc. 3, 2025, págs. 477-492
• Idioma: inglés
• DOI: 10.1007/s13348-024-00441-z
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let I\subset S be a graded ideal of a standard graded polynomial ring S with coefficients in a field K, and let {\text {v}}(I) be the {\text {v}}-number of I. In previous work, we showed that for any graded ideal I\subset S, then {\text {v}}(I^k)=\alpha (I)k+b, for all k\gg 0, where \alpha (I) is the initial degree of I and b is a suitable integer. In the present paper, using polarization, we extend Simon conjecture to any monomial ideal. As a consequence, if Simon conjecture holds, I is a monomial ideal generated in a single degree and all powers of I have linear quotients, then b\in \{-1,0\}. This fact suggests that if I is an equigenerated monomial ideal with linear powers, then {\text {v}}(I^k)=\alpha (I)k-1, for all k\ge 1. We verify this conjecture for monomial ideals with linear powers having {\text {depth}}\,S/I=0, edge ideals with linear resolution, polymatroidal ideals, and Hibi ideals.

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