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19
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
, 2001
"... In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposi ..."
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Cited by 56 (26 self)
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In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As byproducts, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.
Numerical Homotopies to compute generic Points on positive dimensional Algebraic Sets
 Journal of Complexity
, 1999
"... Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the com ..."
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Cited by 50 (24 self)
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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for fourbar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...
Bott’s formula and enumerative geometry
 Journal of the AMS
"... Abstract. We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, GromovWitten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The res ..."
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Cited by 25 (3 self)
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Abstract. We outline a strategy for computing intersection numbers on smooth varieties with torus actions using a residue formula of Bott. As an example, GromovWitten numbers of twisted cubic and elliptic quartic curves on some general complete intersection in projective space are computed. The results are consistent with predictions made from mirror symmetry computations. We also compute degrees of some loci in the linear system of plane curves of degrees less than 10, like those corresponding to sums of powers of linear forms, and curves carrying inscribed polygons. 1.
ErrorCorrecting Codes from HigherDimensional Varieties
 University of Aarhus
, 1998
"... In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric errorcorrecting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from hi ..."
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Cited by 12 (1 self)
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In this paper we use intersection theory to develop methods for obtaining lower bounds on the parameters of algebraic geometric errorcorrecting codes constructed from varieties of arbitrary dimension. The methods are sufficiently general to encompass many of the codes previously constructed from higherdimensional varieties, as well as those coming from curves. And still, the bounds obtained are usually as good as the ones previously known (at least of the same order of magnitude with respect to the size of the ground field). Several examples coming from DeligneLusztig varieties, complete intersections of Hermitian hypersurfaces, and from ruled surfaces (or more generally, projective bundles over a curve) are given.
Segre numbers and hypersurface singularities
 J. Algebraic Geom
, 1999
"... Abstract. We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the ReesBöger theorem, and the ..."
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Cited by 12 (4 self)
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Abstract. We define the Segre numbers of an ideal as a generalization of the multiplicity of an ideal of finite colength. We prove generalizations of various theorems involving the multiplicity of an ideal such as a principle of specialization of integral dependence, the ReesBöger theorem, and the formula for the multiplicity of the product of two ideals. These results are applied to the study of various equisingularity conditions, such as Verdier’s condition W, and conditions Af and Wf. alggeom/9611002 If an ideal I in a Noetherian local ring A has finite colength, then the multiplicity of the ideal is a fundamental invariant of the ideal with many applications in geometry and algebra. Pierre Samuel [S] used it to define the intersection multiplicity of two algebraic sets. David Rees [R] linked the multiplicity of I to its integral
M.: Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces, preprint 2003, 16 pages, appears in the Am
 J. Math
"... Abstract. Let X ⊂ P n be a possibly singular hypersurface of degree d ≤ n, defined over a finite field Fq. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓadic cohomology H i ( ¯ X, Qℓ), for ℓ ̸ = char(Fq), a ..."
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Cited by 8 (7 self)
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Abstract. Let X ⊂ P n be a possibly singular hypersurface of degree d ≤ n, defined over a finite field Fq. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its ℓadic cohomology H i ( ¯ X, Qℓ), for ℓ ̸ = char(Fq), are divisible by q, without using the result on the existence of rational points by Ax and Katz [18]. 1.
Group Completions via Hilbert Schemes
"... Abstract. Let X be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of X belongs to a unique irreducible component HX of the Hilbert scheme of X ×X. Moreover, HX is isomorphic to the “wonderful completion ” of the connected automorphism group of X; in parti ..."
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Cited by 7 (1 self)
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Abstract. Let X be a projective variety, homogeneous under a linear algebraic group. We show that the diagonal of X belongs to a unique irreducible component HX of the Hilbert scheme of X ×X. Moreover, HX is isomorphic to the “wonderful completion ” of the connected automorphism group of X; in particular, HX is nonsingular. We describe explicitly the degenerations of the diagonal in X × X, that is, the points of HX; these subschemes of X ×X are reduced and CohenMacaulay.
The universal regular quotient of the Chow group of points on projective varieties
 Invent. Math
, 1999
"... Let X be a projective variety of dimension n defined over an algebraically closed field k. For X irreducible and nonsingular, Matsusaka [Ma] constructed an abelian variety Alb (X) and a morphism α: X → Alb (X) (called the Albanese variety and mapping respectively), depending on the choice of a base ..."
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Cited by 6 (3 self)
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Let X be a projective variety of dimension n defined over an algebraically closed field k. For X irreducible and nonsingular, Matsusaka [Ma] constructed an abelian variety Alb (X) and a morphism α: X → Alb (X) (called the Albanese variety and mapping respectively), depending on the choice of a basepoint on X, which is universal among the morphisms to abelian varieties (see Lang [La], Serre [Se] for other constructions). Over the field of complex numbers the existence of Alb (X) and α was known before, and has a purely Hodgetheoretic description (see Igusa [I] for the Hodge theoretic construction). Incidentally, the terminology “Albanese variety ” was introduced by A. Weil, for reasons explained in his commentary on the article [1950a] of Volume I of his collected works (see [W]), one of which is that the paper [Alb] of Albanese defines it (for a surface) as a quotient of the group of 0cycles of degree 0 modulo an equivalence relation. Let CH n (X)deg 0 denote the Chow group of 0cycles of degree 0 on X modulo rational equivalence. When X is irreducible and nonsingular, a remarkable feature of the Albanese morphism α is that it factors through a regular homomorphism
The Virtual Poincaré Polynomials of Homogeneous Spaces
 Compositio Math
"... . We factor the virtual Poincar'e polynomial of every homogeneous space G=H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t 2u (t 2 \Gamma 1) r QG=H (t 2 ) for a polynomial QG=H with nonnegative integer coefficients. Moreover, we show that QG=H ( ..."
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Cited by 5 (4 self)
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. We factor the virtual Poincar'e polynomial of every homogeneous space G=H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as t 2u (t 2 \Gamma 1) r QG=H (t 2 ) for a polynomial QG=H with nonnegative integer coefficients. Moreover, we show that QG=H (t 2 ) divides the virtual Poincar'e polynomial of every regular embedding of G=H, if H is connected.