Results 1  10
of
16
A tropical approach to secant dimensions
, 2006
"... Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for th ..."
Abstract

Cited by 23 (1 self)
 Add to MetaCart
Tropical geometry yields good lower bounds, in terms of certain combinatorialpolyhedral optimisation problems, on the dimensions of secant varieties. In particular, it gives an attractive pictorial proof of the theorem of Hirschowitz that all Veronese embeddings of the projective plane except for the quadratic one and the quartic one are nondefective; this proof might be generalisable to cover all Veronese embeddings, whose secant dimensions are known from the groundbreaking but difficult work of Alexander and Hirschowitz. Also, the nondefectiveness of certain Segre embeddings is proved, which cannot be proved with the rook covering argument already known in the literature. Short selfcontained introductions to secant varieties and the required tropical geometry are included.
Decoding AlgebraicGeometric Codes Beyond the ErrorCorrection Bound
, 1998
"... Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with m ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constantrate linear codes over a fixed field F q such that a codeword is efficiently, nonuniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the errorcorrection bound. 1 Introduction Errorcorrecting codes, originally designed to accommodate reliable transmission of information through unreliable ...
Monotone Maps, Sphericity and Bounded Second Eigenvalue
 Journal of Combinatorial Theory, Series B
, 2004
"... We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
We consider monotone embeddings of a nite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on n points can be embedded into l 2 , while, (in a sense to be made precise later), for almost every npoint metric space, every monotone map must be into a space of dimension n) (Lemma 3).
Coset bounds for algebraic geometric codes
, 2008
"... We develop new coset bounds for algebraic geometric codes. The bounds have a natural interpretation as an adversary threshold for algebraic geometric secret sharing schemes and lead to improved bounds for the minimum distance of an AG code. Our bounds improve both floor bounds and order bounds and p ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
We develop new coset bounds for algebraic geometric codes. The bounds have a natural interpretation as an adversary threshold for algebraic geometric secret sharing schemes and lead to improved bounds for the minimum distance of an AG code. Our bounds improve both floor bounds and order bounds and provide for the first time a connection between the two types of bounds.
Combinatorics of the twovariable zeta function.
 LECTURE NOTES IN COMPUT. SCI. 2948
, 2004
"... ..."
(Show Context)
On numerical invariants in algebraic . . .
, 2005
"... A common theme in mathematics is the classification of mathematical objects by assigning numerical invariants to them. There are two ways in which such numerical invariants can appear in relation to computational complexity. On the one hand, mathematical invariants are used in the context of proving ..."
Abstract
 Add to MetaCart
A common theme in mathematics is the classification of mathematical objects by assigning numerical invariants to them. There are two ways in which such numerical invariants can appear in relation to computational complexity. On the one hand, mathematical invariants are used in the context of proving lower complexity bounds: they serve as obstructions to the existence of fast algorithms for solving certain problems. On the other hand, it is the computational complexity of actually computing such invariants that is of interest. The first part of this thesis is concerned with lower bounds for the problems of computing linear and bilinear maps. The invariants used, namely the mean square volume, singular values, and rigidity, belong to linear algebra. One of the main results is a tight lower bound of order Ω(n log n) for the problem of multiplying two polynomials, in the model of bounded coefficient circuits. This lower bound is extended to circuits for which a limited number of unbounded scalar multiplications (help gates) are allowed. The second part is concerned with the complexity of actually computing numerical invariants. The objects of study are two of the most prominent invariants in algebraic geometry and topology: the Euler characteristic and the Hilbert polynomial of complex projective varieties. These problems are studied within the framework of counting complexity classes. It is shown that the problem of computing the Euler characteristic of a complex projective variety is on essentially the same level of difficulty as the problem of counting the number of solutions of a system of polynomial equations. A similar result is proved for the Hilbert polynomials, when the input variety is assumed to be smooth and equidimensional.