See also: ResearchGate, ORCID, Google Scholar, Scopus, Web of Science, arXiv, Mathscinet, zbMATH and GitHub
Current topics:
- Hilbert functions and Betti numbers of quotients of exterior algebras. Characterization of the Hilbert functions of finitely generated modules over an exterior algebra E given the degrees of the possible generators, i.e., of quotients of a fixed free E-module. Moreover, determining of upper bounds for the graded Betti numbers and the graded Bass numbers of such a class of modules via lexicographic submodules.
- Extremal Betti numbers of some classes of ideals. Characterization, values as well as positions, of the extremal Betti numbers of strongly stable ideals (also squarefree or t-spread cases). Algorithmic construction of t-spread strongly stable ideal with a given extremal Betti numbers configuration, if possible.
- Generalized binomial edge ideals. Algebraic and combinatorial properties of generalized binomial edge ideals associated to special classes of graphs.
- Optimal systems of Lie subalgebras. Looking for efficient algorithms to compute optimal systems of finitely generated Lie subalgebras.
- Computational algebra. Developing some packages in CoCoA and Macaulay2 that can be useful in the framework of commutative algebra and algebraic geometry. In particular for deepen the study of the above mentioned topics or approaching some open problems.
Preprints:
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Optimal systems of three and four-dimensional Lie algebrasPreprint.with Francesco Oliveri and Emanuele Sgroi.
Abstract
The optimal systems of three and four-dimensional real Lie algebras by using a new automatic approach via the computer program SymboLie is illustrated. To use such a method, the definition of $p$-family and some related notions are introduced. Furthermore, such results are been analyzed and compared with those reported by Patera and Winternitz.
Papers:
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with Marilena Crupi and Giancarlo Rinaldo.Journal of Algebra and Its Applications, Accepted Papers. doi: 10.1142/S0219498826501227
Abstract
Let $G$ be a simple graph on $n$ vertices and let $J_{G,m}$ be the generalized binomial edge ideal associated to $G$ in the polynomial ring $K[x_{ij}, 1\le i \le m, 1\le j \le n]$. We classify the Cohen--Macaulay generalized binomial edge ideals. Moreover, we study the unmixedness and classify the bipartite and power cycle unmixed ones. -
Optimal Systems of Lie Subalgebras: A Computational Approach journalwith Francesco Oliveri and Emanuele Sgroi.Journal of Geometry and Physics, Vol. 204, 105290 (2024). doi: 10.1016/j.geomphys.2024.105290
Abstract
Lie groups of symmetries of differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of group-invariant solutions. In such cases, it is crucial to obtain a classification of subgroups in order to have an optimal system of inequivalent group-invariant solutions from which all other solutions can be derived by action of the group itself. Since Lie groups are intimately connected to Lie algebras, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite-dimensional Lie algebra relies on the use of inner automorphisms obtained by exponentiating the adjoint group. In this paper, we present a novel effective algorithm able to automatically determine optimal systems of Lie subalgebras of a generic finite-dimensional Lie algebra abstractly assigned by means of its structure constants or realized in terms of matrices or vector fields. The complete algorithm is implemented in the computer algebra system Wolfram Mathematica$^{TM}$; the use of the program is illustrated by means of some meaningful and non-trivial examples. -
Int. Electron. J. Algebra, Vol. 35: 15, pp. 186-216 (2024). doi: 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 Macaulay2 package, TSpreadIdeals, to ease the computation of well-known results about algebraic invariants for $t$-spread ideals. -
Automatic Determination of Optimal Systems of Lie Subalgebras: the Package SymboLie repositorywith Francesco Oliveri.Contemporary Mathematics, Vol. 789, pp. 1–17 (2023).
Abstract
Lie groups of point symmetries of partial differential equations constitute a fundamental tool for constructing group-invariant solutions. The number of subgroups is potentially infinite and so the number of group-invariant solutions. An important goal is a classification in order to have an optimal system of inequivalent group-invariant solutions from which all other solutions can be derived by action of the group itself. In turn, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite-dimensional Lie algebra relies on the use of inner automorphisms. We present a novel effective algorithm that can automatically determine optimal systems of Lie subalgebras of a generic finite-dimensional Lie algebra; here, we limit the analysis to one-dimensional Lie subalgebras, though the same approach still works well for higher dimensional Lie subalgebras. The algorithm is implemented in the computer algebra system Wolfram Mathematica and illustrated by means of some examples. -
with Marilena Crupi and Antonino Ficarra.Internat. J. Algebra Comput., Vol. 32, No. 04, pp. 837-858 (2022). doi: 10.1142/S0218196722500357
Abstract
We study some algebraic invariants of $t$-spread ideals, $t\ge 1$, such as the projective dimension and the Castelnuovo-Mumford regularity, by means of well-known graded resolutions. We state upper bounds for these invariants and, furthermore, we identify a special class of $t$-spread ideals for which such bounds are optimal. -
with Marilena Crupi and Antonino Ficarra.Bull. Math. Soc. Sci. Math. Roumanie (N.S.), Vol. 65(113), No. 01, pp. 13-34 (2022).
Abstract
We study the extremal Betti numbers of the class of $t$-spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $t\in \{1,2\}$. -
A numerical characterization of the extremal Betti numbers of $t$-spread strongly stable ideals journal
arXivwith Marilena Crupi and Antonino Ficarra.Journal of Algebraic Combinatorics, Vol. 5, No. 3, pp. 891-918 (2022). doi: 10.1007/s10801-021-01076-0.Abstract
Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well as positions, of a $t$-spread strongly stable ideal of $S$. Our approach is constructive. Indeed, given some positive integers $a_1,\dots,a_r$ and some pairs of positive integers $(k_1,\ell_1),\dots,(k_r,\ell_r)$, we are able to determine under which conditions there exists a $t$-spread strongly stable ideal $I$ of $S$ with $\beta_{k_i, k_i+\ell_i}(I)=a_i$, $i=1, \ldots, r$, as extremal Betti numbers, and then to construct it. -
ExteriorModules: a package for computing monomial modules over an exterior algebra journal
M2 packagewith Marilena Crupi.Journal of Software for Algebra and Geometry, 11 (1): 71-81, 2021. doi: 10.2140/jsag.2021.11.71.Abstract
Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. We present a Macaulay2 package to manage some classes of monomial submodules of $F$. The package is an extension of the one on monomial ideals (ExteriorIdeals) and contains some algorithms for computing stable, strongly stable and lexicograhic $E$-submodules of $F$. Such a package also includes some methods to check whether a sequence of nonnegative integers is the Hilbert function of a graded $E$-module of the form $F/M$, with $M$ graded submodule of $F$. Moreover, if $H_{F/M}$ is the Hilbert function of a graded $E$-module $F/M$, some routines are able to compute the unique lexicograhic submodule $L$ of $F$ such that $H_{F/M} = H_{F/L}$. -
On the extremal Betti numbers of squarefree monomial ideals journalwith Marilena Crupi.Int. Electron. J. Algebra, 30: 7, 2021.
Abstract
Let $K$ be a field and $S=K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We discuss the behaviour of the extremal Betti numbers of the class of squarefree strongly stable ideals. More precisely, we give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of such a class of squarefree monomial ideals. -
A generalization of Kruskal-Katona's theorem journalwith Marilena Crupi.An. Ştiinţ. Univ. ''Ovidius'' Constanţa Ser. Mat., 28 (2): 35–51, 2020. doi: 10.2478/auom-2020-0018.
Abstract
Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. We characterize the Hilbert functions of graded $E$-modules of the type $F/M$, with $M$ graded submodule of $F$. The existence of a unique lexicographic submodule of $F$ with the same Hilbert function as $M$ plays a crucial role. -
Graded algebras: theoretical and computational aspects urlPhD Thesis, University of Catania, 2020.
Abstract
In this dissertation we study by a computational approach Hilbert functions and minimal graded free resolutions of finitely generated graded modules over two significant graded $K$-algebras, $K$ being a field. More precisely, if $E$ is the exterior algebra of a finite dimensional $K$-vector space and $F$ is a finitely generated graded free $E$-module with a homogeneous basis, we characterize the Hilbert functions of graded $E$-modules of the type $F/M$, with $M$ graded submodule of $F$, via the unique lexicographic submodule of $F$ having the same Hilbert function as $M$. Furthermore, we study projective and injective resolutions over $E$. In particular, we give upper bounds for the graded Betti numbers and the graded Bass numbers of classes of $E$-modules. Moreover, we give a criterion to determine the extremal Betti numbers of a special class of monomial ideals of a standard polynomial ring $S$ known as the $t$-spread strongly stable ideals, where $t$ is an integer $\ge 0$. We are able to find a complete numerical characterization (positions as well as values) for the case $t=0$ and $t=1$. Instead, for the case $t=2$ we determine the structure of the $t$-spread strongly stable ideals with the maximal number of extremal Betti numbers. The approach to these topics is mainly computational because of the algorithmic nature of the topic themselves. Finally, we present some packages in order to work and manipulate specific objects in both contexts. -
Hilbert functions of graded modules over exterior algebras: an algorithmic approach journalwith Marilena Crupi.Int. Electron. J. Algebra, 27: 271–287, 2020. doi: 10.24330/ieja.663094.
Abstract
Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. Given the Hilbert function of a graded $E$-module of the type $F/M$, with $M$ graded submodule of $F$, the existence of the unique lexicographic submodule of $F$ with the same Hilbert function as $M$ is proved by a new algorithmic approach. Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free $E$-module only via the combinatorial Kruskal-Katona's theorem. -
Extremal Betti numbers of t-spread strongly stable ideals journalwith Marilena Crupi.Mathematics, 7 (695), 2019. doi: 10.3390/math7080695.
Abstract
Let $K$ be a field and let $S = K[x_1,\dots,x_n]$ be a polynomial ring over $K$. We analyze the extremal Betti numbers of special squarefree monomial ideals of $S$ known as the $t$-spread strongly stable ideals, where $t$ is an integer $\ge 1$. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the $t$-spread strongly stable ideals with the maximal number of extremal Betti numbers when $t=2$. -
Minimal resolutions of graded modules over an exterior algebra journalwith Marilena Crupi.Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 97 (1): A5, 2019. doi: 10.1478/AAPP.971A5.
Abstract
Let $K$ be a field, $E$ the exterior algebra of a $n$-dimensional $K$-vector space $V$. We study projective and injective resolutions over $E$. More precisely, given a category $\mathcal{M}$ of finitely generated $\mathbb{Z}$-graded left and right $E$-modules, we give upper bounds for the graded Betti numbers and the graded Bass numbers of classes of modules in $\mathcal{M}$. -
Computation of graded ideals with given extremal Betti numbers in a polynomial ring journal
CoCoA packagewith Marilena Crupi.Journal of Symbolic Computation, 93: 120-132, 2019. doi: 10.1016/j.jsc.2018.04.019.Abstract
Consider a polynomial ring in a finite number of variables over a field of characteristic $0$. We implement in CoCoA some algorithms in order to easy compute graded ideals of this ring with given extremal Betti numbers (positions as well as values). More precisely, we develop a package for determining the conditions under which, given two positive integers $n, r$, $1\le r \le n-1$, there exists a graded ideal of a polynomial ring in $n$ variables with $r$ extremal Betti numbers in the given position. An algorithm to check whether an $r$-tuple of positive integers represents the admissible values of the $r$ extremal Betti numbers is also described. An example in order to show how the package works is also presented. -
Bounds for the Betti numbers of graded modules with given Hilbert function in an exterior algebra via lexicographic modules journalwith Marilena Crupi.Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 61 (109) (3): 237-253, 2018.
Abstract
Let $K$ be a field, $V$ a finite dimensional $K$-vector space, $E$ the exterior algebra of $V$, and $F$ a finitely generated graded free $E$-module with all basis elements of the same degree. We prove that given any graded submodule $M$ of $F$, there exists a unique lexicographic submodule $L$ of $F$ such that $H_{F/L}=H_{F/M}$. As a consequence, we are able to describe the possible Hilbert functions of graded $E$-modules of the type $F/M$. Finally, we state that the lexicographic submodules of $F$ give the maximal Betti numbers among all the graded submodules of $F$ with the same Hilbert function. -
with Marilena Crupi.Journal of Software for Algebra and Geometry, 8 (1): 71-79, 2018. doi: 10.2140/jsag.2018.8.71.
Abstract
Let $K$ be a field, $V$ a $K$-vector space with basis $e_1,\ldots,e_n$, and $E$ the exterior algebra of $V$. We introduce a Macaulay2 package that allows one to deal with classes of monomial ideals in $E$. More precisely, we implement in Macaulay2 some algorithms in order to easily compute stable, strongly stable and lexsegment ideals in $E$. Moreover, an algorithm to check whether an $(n+1)$-tuple $(1, h_1, \ldots, h_n)$ ($h_1 \le n= \dim_K V$) of non-negative integers is the Hilbert function of a graded $K$-algebra of the form $E/I$, with $I$ graded ideal of $E$, is given. In particular, if $H_{E/I}$ is the Hilbert function of a graded $K$-algebra $E/I$, the package is able to construct the unique lexsegment ideal $I^\mathrm{lex}$ such that $H_{E/I} = H_{E/I^\mathrm{lex}}$.