We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a represen- tation... more
We present a new technique, based on polynomial continuation, for solving sys- tems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a represen- tation of the solution set that can be extended to the next stage until nally obtaining the solution set for the entire system.
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This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X be an algebraic set deflned by a system of homogeneous multivariate polynomials with coe-cients in C. Let C be a union of reduced,... more
This article presents several numerical algorithms for computa- tions in sheaf cohomology. Let X be an algebraic set deflned by a system of homogeneous multivariate polynomials with coe-cients in C. Let C be a union of reduced, irreducible pure-dimensional curve components of X. The flrst algorithm computes the dimension of the flrst cohomology of any twist of the ideal sheaf
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Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore key in most algorithms in the emerging eld of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been... more
Path tracking is the fundamental computational tool in homotopy con- tinuation and is therefore key in most algorithms in the emerging eld of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. This article considers two ways
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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 com-puting the dimensions of secant... more
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 com-puting 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.
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Numerical di-culties encountered when following paths using methods such as homotopy continuation may be overcome by combining adaptive stepsize and adaptive multiprecision. In the paper Adaptive multipreci- sion path tracking (1),... more
Numerical di-culties encountered when following paths using methods such as homotopy continuation may be overcome by combining adaptive stepsize and adaptive multiprecision. In the paper Adaptive multipreci- sion path tracking (1), precision and stepsize are adapted separately. This can lead to suboptimal performance and even failure in certain circum- stances. This paper presents a strategy for adjusting precision and stepsize
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... Corollary 2.8. Leí X be a» vidimevisio»al smooth projeclive variety avid leí 8 be a lejet spa»vied vector buvidie of ravile r ovi X. Thevi 1. c~,(S) - ..c(S) =leQ) - . -(4) forO =~ ~ r,i1 + ---+ij = vi; 2. s~1(S) -sé(S)... more
... Corollary 2.8. Leí X be a» vidimevisio»al smooth projeclive variety avid leí 8 be a lejet spa»vied vector buvidie of ravile r ovi X. Thevi 1. c~,(S) - ..c(S) =leQ) - . -(4) forO =~ ~ r,i1 + ---+ij = vi; 2. s~1(S) -sé(S) =len(r+:i~i) . . - (r+itl) forO =¿~ ~ r, lí + - - - + tt = vi. ...
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How could only one year's worth of download statistics be a fair measure of use? Upon reflection, the second author of this article uncovered an even more serious flaw: the downloads had not been adjusted for the journals'... more
How could only one year's worth of download statistics be a fair measure of use? Upon reflection, the second author of this article uncovered an even more serious flaw: the downloads had not been adjusted for the journals' half-lives. Because the electronic runs are short ...
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Page 1. ON K-SPANNEDNESS FOR PROJECTIVE SURFACES Mauro Beltrametti Dipartimento di Matematica Via LB Alberti 4, 1-16132 Genova, Italy. Andrew J. Sommese Department of Mathematics, University of Notre Dame Notre Dame, Indiana 46556, USA ...
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this paper we present two results on branched coverings of Grassmannians.Throughout this introduction, let G := Gr (r; n) denote the Grassmannian of r-dimensional complexvector subspaces of C
We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and... more
We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two form.