Computational methods for t-spread monomial ideals

Year 2024, , 186 - 216, 09.01.2024

Luca Amata

https://doi.org/10.24330/ieja.1402973

Abstract

Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ a standard polynomial ring over $K$.
In this paper, we give new combinatorial algorithms to compute the smallest $t$-spread lexicographic set and the smallest $t$-spread strongly stable set containing a given set of $t$-spread monomials of $S$.
Some technical tools allowing to compute the cardinality of $t$-spread strongly stable sets avoiding their construction are also presented.
Such functions are also implemented in a \emph{Macaulay2} package, \texttt{TSpreadIdeals}, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals.

Keywords

Squarefree monomial ideal, $t$-spread ideal, strongly stable ideal, lexicographic ideal, extremal Betti number, computational algebra

References

• L. Amata, Graded Algebras: Theoretical and Computational Aspects, Doctoral Thesis, University of Catania, 2020.
• L. Amata and M. Crupi, Extremal Betti numbers of $t$-spread strongly stable ideals, Mathematics, {7}(8) (2019), 695 (16 pp).
• L. Amata and M. Crupi, On the extremal Betti numbers of squarefree monomial ideals, Int. Electron. J. Algebra, {30} (2021), 168-202.
• L. Amata, M. Crupi and A. Ficarra, Upper bounds for extremal Betti numbers of $t$-spread strongly stable ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 65(113)(1) (2022), 13-34.
• L. Amata, M. Crupi and A. Ficarra, Projective dimension and Castelnuovo-Mumford regularity of $t$-spread ideals, Internat. J. Algebra Comput., 32(4) (2022), 837-858.
• L. Amata, A. Ficarra and M. Crupi, A numerical characterization of the extremal Betti numbers of $t$-spread strongly stable ideals, J. Algebraic Combin., 55(3) (2022), 891-918.
• C. Andrei-Ciobanu, V. Ene and B. Lajmiri, Powers of $t$-spread principal Borel ideals, Arch. Math. (Basel), {112}(6) (2019), 587-597.
• C. Andrei-Ciobanu, Kruskal-Katona Theorem for $t$-spread strongly stable ideals, Bull. Math. Soc. Sci. Math. Roumanie ({N.S.}), {62(110)(2)} (2019), 107-122.
• A. Aramova, J. Herzog and T. Hibi, Squarefree lexsegment ideals, Math. Z., {228}(2) (1998), 353-378.
• A. Aramova, J. Herzog and T. Hibi, Shifting operations and graded Betti numbers, J. Algebraic Combin., {12}(3) (2000), 207-222.
• D. Bayer, H. Charalambous and S. Popescu, Extremal Betti numbers and applications to monomial ideals, J. Algebra, {221}(2) (1999), 497-512.
• R. Dinu, J. Herzog and A. A. Qureshi, Restricted classes of veronese type ideals and algebras, Internat. J. Algebra Comput., {31}(1) (2021), 173-197.
• D. Eisenbud, Commutative Algebra, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995.
• V. Ene, J. Herzog and A. A. Qureshi, $T$-spread strongly stable monomial ideals, Comm. Algebra, 47(12) (2019), 5303-5316.
• D. R. Grayson and M. E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www2.macaulay2.com.
• J. Herzog and T. Hibi, Monomial Ideals, Grad. Texts in Math., 260, Springer-Verlag, London, 2011.
• E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Grad. Texts in Math., 227, Springer-Verlag, New York, 2005.