Aldo Conca

An algebraic characterization of injectivity in phase retrieval

Abstract A complex frame is a collection of vectors that span C M and define measurements, called intensity measurements, on vectors in C M . In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4 M − 4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

M-Sequences, Graph Ideals, and Ladder Ideals of Linear Type☆

In this paper we study monomial ideals and ladder determinantal ideals of linear type and their blow-up algebras. Our main tools are Grobner bases and Sagbi bases deformations and the notion of M-sequence of monomials. w x Let R s K X be a polynomial ring over a field K equipped with a monomial order t . Let I be an ideal of R generated by polynomials f , . . . , f . Consider the presentation 1 s w x w x w x c : R T s R T , . . . , T a R I s R f t , . . . , f t Ž . 1 s i s

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s ≤ 2n points in general linear position in Pn, and the Koszul property of the ‘generic’ Artinian Gorenstein algebra of socle degree 3.

Reduction numbers and initial ideals

The reduction number r(A) of a standard graded algebra A is the least integer k such that there exists a minimal reduction J of the homogeneous maximal ideal m of A such that Jm k = m k+1 . Vasconcelos conjectured that r(R/I) < r(R/in(I)) where in(I) is the initial ideal of an ideal I in a polynomial ring R with respect to a term order. The goal of this note is to prove the conjecture.

Ladder determinantal rings

In this paper we show that ladder determinantal rings are normal. In the case of a ladder determinantal ring associated with a one-sided ladder, we compute the divisor class group, the canonical class, and we obtain a characterization of the Gorensteinness in terms of the shape of the ladder.

Koszul homology and extremal properties of Gin and Lex

For every homogeneous ideal I in a polynomial ring R and for every p < dim R we consider the Koszul homology H i (p, R/I) with respect to a sequence of p of generic linear forms. The Koszul-Betti number β ijp (R/I) is, by definition, the dimension of the degree j part of H i (p, R/I). In characteristic 0, we show that the Koszul-Betti numbers of any ideal I are bounded above by those of the gin-revlex Gin(I) of I and also by those of the Lex-segment Lex(I) of I. We show that β ijp (R/I) = β ijp (R/Gin(I)) iff I is componentwise linear and that and β ijp (R/I) = β ijp (R/Lex(I)) iff I is Gotzmann. We also investigate the set Gins(I) of all the gin of I and show that the Koszul-Betti numbers of any ideal in Gins(I) are bounded below by those of the gin-revlex of I. On the other hand, we present examples showing that in general there is no J is Gins(I) such that the Koszul-Betti numbers of any ideal in Gins(I) are bounded above by those of J.

Gröbner Bases and Determinantal Ideals

We give an introduction to the theory of determinantal ideals and rings, their Grobner bases, initial ideals and algebras, respectively. The approach is based on the straightening law and the Knuth-Robinson-Schensted correspondence. The article contains a section treating the basic results about the passage to initial ideals and algebras.

Generic initial ideals of points and curves

Let I be the defining ideal of a smooth complete intersection space curve C with defining equations of degrees a and b. We use the partial elimination ideals introduced by Mark Green to show that the lexicographic generic initial ideal of I has Castelnuovo-Mumford regularity 1+ab(a-1)(b-1)/2 with the exception of the case a=b=2, where the regularity is 4. Note that ab(a-1)(b-1)/2 is exactly the number of singular points of a general projection of C to the plane. Additionally, we show that for any term ordering @t, the generic initial ideal of a generic set of points in P^r is a @t-segment ideal.

Hilbert-Kunz function of monomial ideals and binomial hypersurfaces

The aim of this note is to determine the Hilbert-Kunz functions of rings defined by monomial ideals and of rings defined by a single binomial equationXa−Xb with gcd(Xa, Xb)=1.

REGULAR SEQUENCES OF SYMMETRIC POLYNOMIALS

Denote by p_k the k-th power sum symmetric polynomial n variables. The interpretation of the q-analogue of the binomial coefficient as Hilbert function leads us to discover that n consecutive power sums in n variables form a regular sequence. We consider then the following problem: describe the subsets n powersums forming a regular sequence. A necessary condition is that n! divides the product of the degrees of the elements. To find an easily verifiable sufficient condition turns out to be surprisingly difficult already in 3 variables. Given positive integers a<b<c with GCD(a,b,c)=1, we conjecture that p_a, p_b, p_c is a regular sequence for n=3 if and only if 6 divides abc. We provide evidence for the conjecture by proving it in several special instances.