Aron Simis

On a conjecture of Vasconcelos via Sylvester forms

Journal of Symbolic Computation, 2016

DESCRIPTION This is an updated version of &quot;On a conjecture of Vasconcelos&quot;, by ... more

Rees algebras and elimination algebras A TALK AT THE UNIVERSITY OF VALLADOLID

The author thanks the Departamento de Algebra of the University of Valladolid and CNPq (Brazil) f... more The author thanks the Departamento de Algebra of the University of Valladolid and CNPq (Brazil) for providing the resources for this visit. Contents 1 Introduction 1 2 Magic words 3 2.1 The first magic word . Abstract The talk surveys results on typical constructions borrowed from Intersection Theory, by mainly focusing on their algebraic side and the problems issuing thereof. One also gives an overview of open problems and some tools to deal with them.

Polar syzygies of monomials and their graph-theoretic side

Polar syzygies of monomials and their graph-theoretic side or How to pretend that a graph is an a... more

Linear Syzygies and Ideals of Linear Type (In collaboration with F. Russo)

I. ALGEBRAIC PRELIMINARIES II. IDEAL THEORETIC METHOD for RATIONAL MAPS III. Cremona transformations

On implicitization and equations in low dimension

This presentation is around the eleimination aspects of parameters generating an ideal which is a... more This presentation is around the eleimination aspects of parameters generating an ideal which is an almost complete intersection of finite colength in a polynomial ring over a field. There are natural and technical reasons to focus on almost complete intersections. First, a good deal of elimination theory is intertwined with birationality questions. Now, a regular sequence of forms of fixed degree � 2 never defines a birational map. Thus, the first relevant case is that of an almost complete intersection.

Monomial Cremona Maps and Its Combinatorics

We tackle the following questions: Classification of monomial Cremona maps of degree 2 in any num... more We tackle the following questions: Classification of monomial Cremona maps of degree 2 in any number of variables The structure of the inverse map to a monomial Cremona map of degree 2 The role of Hilbert bases in monomial Cremona maps. We focus on the relevant combinatorial data, thus retracting from the classical theory of Cremona transformations in characteristic zero.

BIRATIONAL MAPS : An overview of ideal theoretic tools or how SYZYGIES help to deal with indeterminacy loci

Topics: The hard times, Italian–German–Italian and all that jazz Is “classical” modern? Syzygies ... more

IMPLICITIZATION of de Jonquìères like parametrizations

We tackle the following aspects: 1. Homological preliminaries of a de Jonqui ´eres parametrizatio... more We tackle the following aspects: 1. Homological preliminaries of a de Jonqui ´eres parametrization 2. The intervenience of the syzygetic polynomial 3. The role of the inversion factor of a birational parametrization 4. Formulas and inequalities for the degree of the implicit equation 5. Relation to the monoid parametrization. (Joint work with S. H. Hassanzadeh)

The role and power of mathematics (Highlights of a Public Lecture)

Topics in rees algebras of special ideals

Lecture Notes in Mathematics, 1990

... ARON SIMIS* 1. Introduction ... (2.5) Definition. Let A be a graded ordinal Hodge algebra on ... more

Computational aspects of jacobian matrices

Foundations of Computational Mathematics, 1997

The underlying theme of this talk is the jacobian matrix of a (finite) set of multivariate polyno... more The underlying theme of this talk is the jacobian matrix of a (finite) set of multivariate polynomials f = {l1…-, lm} ⊂ R = k[X] = k[X1…,Xd]. Here k stands for a field, eventually of characteristic zero. Jacobian matrices are classical, useful in almost all branches of mathematics and there is a famous conjecture that bears the name! Is this enough reason to make them a central theme in a talk? Perhaps. The full justification here would go far beyond our modest purpose which is, namely, to convey their theoretical and computational usefulness in everyday commutative algebra and algebraic geometry.

Symbolic powers of perfect ideals of codimension 2 and birational maps

Journal of Algebra, 2014

This work is about symbolic powers of codimension two perfect ideals in a standard polynomial rin... more This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal. A full description of the corresponding symbolic Rees algebra is given in some cases. An application is an affirmative solution of a conjecture of Eisenbud-Mazur in [11, Section 2].

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On Birational Maps and Jacobian Matrices

Compositio Mathematica - COMPOS MATH, 2001

One is concerned with Cremona-like transformations, i.e., rational maps from Pn to Pm that are bi... more One is concerned with Cremona-like transformations, i.e., rational maps from Pn to Pm that are birational onto the image Y ? Pm and, moreover, the inverse map from Y to Pn lifts to Pm. We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.

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Projective Moduli and Maximal Spectra of Certain Quotient Rings

Transactions of the American Mathematical Society, 1972

Projective modulus, maximal spectrum, families of prime ideals of Towber type, maximal type, unim... more

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On the arithmetic and homology of algebras of linear type

Transactions of the American Mathematical Society, 1984

Three modifications of the symmetric algebra of a module are introduced and their arithmetical an... more Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.

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Cremona maps defined by monomials

Journal of Pure and Applied Algebra, 2012

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer ... more Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones. The latter suggests that facets of monomial Cremona theory may be NP-hard.

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The Integral Closure of Subrings Associated to Graphs

Journal of Algebra, 1998

Let G be a graph on the vertex set V, and R s k V a polynomial ring over a field k. We give a ver... more

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Projective modules of certain rings and the existence of cyclic basis

Journal of Algebra, 1971

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Arithmetic of normal Rees algebras

Journal of Algebra, 1991

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Koszul homology and its syzygy-theoretic part

Journal of Algebra, 1978

To parody the preface of a famous book, this work is a side product of an as yet unborn research.... more To parody the preface of a famous book, this work is a side product of an as yet unborn research. Namely, in trying to tell singularities of a curve from the Koszul homology on the generators of its Jacobian ideal, we were led to examine the so-called syzygy-theoretic part of the Koszul homology. In the first section of this work we revisit somewhat more carefully the notion of "syzygietic" ideal introduced in [5], h aving in mind a different viewpoint. We show that the concept of I?r-faithfulness is well-defined for an ideal in a commutative ring (cf. Section 1 for the definition of IIk-faithfulness). In the second section the question of the vanishing of the syzygy-theoretic part is tied up with the R/I-module I/Id, where I is the ideal in consideration. As a consequence we obtain that, in certain situations, the vanishing of the syzygy-theoretic part is very close to the vanishing of the Koszul homology itself (cf. Propositions 2.2 and 2.3). Section 3 is devoted to the discussion of I&-faithfulness for doubly generated ideals, while the last section makes a brief incursion in the world of ideals generated by three elements. A more systematic treatment will appear elsewhere.

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