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15
On the rank of a binary form
"... Abstract. We describe in the space of binary forms of degree d the strata of forms having constant rank. We also give a simple algorithm to determine the rank of a given form. 1. ..."
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Abstract. We describe in the space of binary forms of degree d the strata of forms having constant rank. We also give a simple algorithm to determine the rank of a given form. 1.
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.
ON THE RANKS AND BORDER RANKS OF SYMMETRIC TENSORS
"... Abstract. Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by ..."
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Cited by 11 (0 self)
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Abstract. Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent. 1.
Subspace arrangements defined by products of linear forms
 In Journal of the London Math. Society
, 2005
"... Abstract. We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatoria ..."
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Cited by 10 (2 self)
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Abstract. We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of points in P 2 and lines in P 3. 1.
The Number of Embeddings of Minimally Rigid Graphs
 GEOMETRY © 2003 SPRINGERVERLAG NEW YORK INC.
, 2003
"... Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, mo ..."
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Cited by 7 (2 self)
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Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most � � 2n−4 n ≈ 4. We also exhibit several families which realize lower bounds n−2 of the order of 2n,2.21n and 2.28n. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley–Menger variety CM2,n (C) ⊂ P n ( 2)−1(C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM2,n) zerodimensional components, and one finds this degree to be D2,n � � = 1 2n−4. The lower bounds are related to inductive constructions of minimally rigid graphs 2 n−2 via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of 2D3,n = (2n−3 /(n − 2)) � � 2n−6 for the number of spatial embeddings n−3 with generic edge lengths of the 1skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the nonEuclidean case.
Commuting pairs and triples of matrices and related varieties
, 2000
"... In this note, we show that the set of all commuting dtuples of commuting n × n matrices that are contained is an ndimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting ..."
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In this note, we show that the set of all commuting dtuples of commuting n × n matrices that are contained is an ndimensional commutative algebra is a closed set, and therefore, Gerstenhaber’s theorem on commuting pairs of matrices is a consequence of the irreducibility of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of ndimensional commutative subalgebras of Mn(F), and show that it is irreducible of dimension n 2 − n for n ≤ 4, but reducible, of dimension greater than n 2 − n for n ≥ 7.
Point configurations and Cayley–Menger varieties, manuscript,http://arxiv.org/abs/ math/0207110
, 2002
"... Equivalence classes of npoint configurations in Euclidean, Hermitian, and quaternionic spaces are related, respectively, to classical determinantal varieties of symmetric, general, and skewsymmetric bilinear forms. CayleyMenger varieties arise in the Euclidean case, and have relevance for mechani ..."
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Cited by 4 (2 self)
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Equivalence classes of npoint configurations in Euclidean, Hermitian, and quaternionic spaces are related, respectively, to classical determinantal varieties of symmetric, general, and skewsymmetric bilinear forms. CayleyMenger varieties arise in the Euclidean case, and have relevance for mechanical linkages, polygon spaces and rigidity theory. Applications include upper bounds for realizations of planar Laman graphs with prescribed edgelengths and examples of special Lagrangians in CalabiYau manifolds. Introduction. We are concerned, initially, with configurations of n labeled (or ordered) points in the Euclidean space R d, up to equivalence under congruence and similarity (rescaling). We require at least two points to be distinct and denote by Cn(R d) the resulting configuration space, made of such equivalence classes.
On tangents to quadric surfaces
, 2004
"... We study the variety of common tangents for up to four quadric surfaces in projective threespace, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces w ..."
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Cited by 3 (2 self)
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We study the variety of common tangents for up to four quadric surfaces in projective threespace, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean threespace, and result in a complete answer to the question over the reals: “When do four spheres allow infinitely many common tangents?”.
Implicitization of a General Union of Parametric Varieties
, 2001
"... this paper we consider the implicitization of a general parametric variety V and we introduce an alternative method that reconducts the implicitization of V to the computation of equations that dene a nite set of points of V . This allows us to use some known algorithms that construct minimal genera ..."
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this paper we consider the implicitization of a general parametric variety V and we introduce an alternative method that reconducts the implicitization of V to the computation of equations that dene a nite set of points of V . This allows us to use some known algorithms that construct minimal generators and Hilbert functions of ideals of projective points in a time polynomial in the number of points and in the dimension of the projective space ( Orecchia (1991); Ramella (1994), Marinari et al. (1993); Cio (1999)). We show that the implementation of this new algorithm of implicitization is more ecient than the ones contained in CoCoA3.7 and Singular1.2. Moreover our approach can be applied with no signicant changes to the implicitization of a general union V = S q i=1 V i of projective parametric varieties. In this case it works much more eciently with respect to the other procedures that rst implicitize all the varieties V i , and then compute the equations dening V by ideal intersection. Let IP n denote the projective ndimensional space over an algebraically closed eld k. Let V = S q i=1 V i IP n be the union of parametric varieties V i parametrized by maps i : IP m i ! IP n , m i < n for any i, given by homogeneous polynomials of the same degree 07477171/90/000000 + 00 $03.00/0 c 2001 Academic Press Limited 2 F. Orecchia r i . Let I(V ) R = k[X 0 ; :::; Xn ] be the ideal of V . Fix an integer d, and let P (i) 1 ; :::; P (i) N i be N i = dr i +m i m i be points of IP m i in generic position such that Q (i) j = i (P (i) j ) form a set T of N = P q i=1 N i points of IP n . Let f be a homogeneous polynomial in n + 1 variables with degree deg f d. The equation f = 0 denes V (i.e. f 2 I(V )) if and only if it denes T (...
A Review of: Computational algebraic geometry, by H. Schenck, London Mathematical Society
, 2004
"... Algebraic geometry is a powerful combination of algebra and geometry, with a rich history and many applications, both theoretical and practical. It also has an intimidating reputation. Hence it is important to have a variety of introductory texts at different levels of sophistication. We will see th ..."
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Algebraic geometry is a powerful combination of algebra and geometry, with a rich history and many applications, both theoretical and practical. It also has an intimidating reputation. Hence it is important to have a variety of introductory texts at different levels of sophistication. We will see that Schenck’s book offers an interesting path into this wonderful subject.