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33
Geometry and the complexity of matrix multiplication
, 2007
"... Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, ..."
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Cited by 15 (1 self)
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Abstract. We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii) to motivate researchers to work on these questions, and (iii) to point out relations with more general problems in geometry. The key geometric objects for our study are the secant varieties of Segre varieties. We explain how these varieties are also useful for algebraic statistics, the study of phylogenetic invariants, and quantum computing.
Singularities of linear systems and the Waring problem. Preprint math.AG/0406288 (2004). Claudio Fontanari Università degli Studi di Trento Dipartimento di Matematica Via Sommarive 14 38050 Povo (Trento) Italy email: fontanar@science.unitn.it
"... Edward Waring stated in 1770 that every integer is a sum of at most 9 positive integral cubes, also a sum of at most 19 biquadrates and so on. Later on Jacobi and others considered the problem to find all the decompositions of a given number into the least number of powers, [Di]. In this paper I am ..."
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Cited by 12 (1 self)
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Edward Waring stated in 1770 that every integer is a sum of at most 9 positive integral cubes, also a sum of at most 19 biquadrates and so on. Later on Jacobi and others considered the problem to find all the decompositions of a given number into the least number of powers, [Di]. In this paper I am concerned with a similar
Topics on Interpolation Problems in Algebraic Geometry
, 2004
"... These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results ..."
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Cited by 9 (3 self)
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These are notes of the lectures given by the authors during the school/workshop “Polynomial Interpolation and Projective Embeddings”. We mainly focus our attention on the planar case and on the Segre and HarbourneHirschowitz Conjectures. We discuss the state of the art introducing several results and different techniques.
Reducing the number of variables of a polynomial
 Algebraic geometry and geometric modeling
, 2005
"... In this paper, we consider two basic questions about presenting a homogeneous polynomial f: how many variables are needed for presenting f? How can one find a presentation of f involving as few variables as possible? We give a complete answer to both questions, determining the minimal number of vari ..."
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Cited by 7 (0 self)
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In this paper, we consider two basic questions about presenting a homogeneous polynomial f: how many variables are needed for presenting f? How can one find a presentation of f involving as few variables as possible? We give a complete answer to both questions, determining the minimal number of variables needed, Ness(f), and describing these variables through their linear span, EssVar(f). Our results give rise to effective algorithms which we implemented in the computer algebra system CoCoA [CoC04].
A limit linear series moduli scheme
"... Abstract. We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the EisenbudHarris theory, and shows promise for generalization to higherdimensional varieties and higherrank vector bundles. We also give a result on lifting ..."
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Cited by 6 (0 self)
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Abstract. We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the EisenbudHarris theory, and shows promise for generalization to higherdimensional varieties and higherrank vector bundles. We also give a result on lifting linear series from characteristic p to characteristic 0. In an appendix, in order to obtain the necessary dimensional lower bounds on our limit linear series scheme we develop a theory of “linked Grassmannians; ” these are schemes parametrizing subbundles of a sequence of vector bundles which map into one another under fixed maps of the ambient bundles. 1.
Curves having one place at infinity and linear systems on rational surfaces
 J. Pure Appl. Algebra
"... Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide ..."
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Cited by 4 (4 self)
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Abstract. Denoting by Ld(m0, m1,..., mr) the linear system of plane curves passing through r + 1 generic points p0, p1,..., pr of the projective plane with multiplicity mi (or larger) at each pi, we prove the HarbourneHirschowitz Conjecture for linear systems Ld(m0, m1,..., mr) determined by a wide family of systems of multiplicities m = (mi) r i=0 and arbitrary degree d. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system m and we give its exact value when m is in the above family. To do that, we prove an H 1vanishing theorem for line bundles on surfaces associated with some pencils “at infinity”. 1.
Bivariate Hermite interpolation via computer algebra and algebraic geometry techniques
, 2002
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Base loci of linear systems and the Waring problem
 Proc. Amer. Math. Soc
"... The Waring problem for forms is the quest for an additive decomposition of homogeneous polynomials into powers of linear ones. The subject has been widely considered in old times, [Sy], [Hi], [Ri] and [Pa], with special regards to the existence of a unique decomposition of this type. The following, ..."
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Cited by 3 (0 self)
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The Waring problem for forms is the quest for an additive decomposition of homogeneous polynomials into powers of linear ones. The subject has been widely considered in old times, [Sy], [Hi], [Ri] and [Pa], with special regards to the existence of a unique decomposition of this type. The following, see [RS], was the state of
Classifying Hilbert functions of fat point subschemes
"... Abstract. The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary charact ..."
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Cited by 3 (1 self)
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Abstract. The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [H2] and [H3] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p1,..., pr and p ′ 1,..., p ′ r have the same configuration type if for all choices of nonnegative integers mi, Z = m1p1 + · · · + mrpr and Z ′ = m1p ′ 1 + · · · + mrp ′ r have the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [GuH] for the cases r ≤ 6) or that the points pi lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients mi determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m1p1 + · · · + mrpr. We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function. 1.
Applications of a Numerical Version of Terracini’s Lemma for Secants and Joins
, 2006
"... This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini’s lemma, can be used to produce a high probability algorithm for computing the dimensions of secant and join varieties. The use of numerical methods allows applications ..."
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This paper illustrates how methods such as homotopy continuation and monodromy, when combined with a numerical version of Terracini’s lemma, can be used to produce a high probability algorithm for computing the dimensions of secant and join varieties. The use of numerical methods allows applications to problems that are difficult to handle by purely symbolic algorithms.