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49
WHICH POWERS OF HOLOMORPHIC FUNCTIONS ARE INTEGRABLE?
, 2008
"... The aim of this lecture is to investigate the following, rather elementary, problem: Question 1. Let f(z1,..., zn) be a holomorphic function on an open set U ⊂ C n. For which t ∈ R is f  t locally integrable? The positive values of t pose no problems, for these f  t is even continuous. If f is n ..."
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The aim of this lecture is to investigate the following, rather elementary, problem: Question 1. Let f(z1,..., zn) be a holomorphic function on an open set U ⊂ C n. For which t ∈ R is f  t locally integrable? The positive values of t pose no problems, for these f  t is even continuous. If f is nowhere zero on U then again f  t is continuous for any t ∈ R. Thus the question is only interesting near the zeros of f and for negative values of t. More generally, if h is an invertible function then f  t locally integrable iff fh  t is locally integrable. Thus the answer to the question depends only on the hypersurface (f = 0) but not on the actual equation. (A hypersurface (f = 0) is not just the set where f vanishes. One must also remember the vanishing multiplicity for each irreducible component.) It is traditional to change the question a little and work with s = −t/2 instead. Thus we fix a point p ∈ U and study the values s such that f  −s is L 2 in a neighborhood of p. It is not hard to see that there is a largest value s0 (depending on f and p) such that f  −s is L 2 in a neighborhood of p for s < s0 but not L 2 for
Computation of unirational fields
 J. Symbolic Comput
"... In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Gröbner bases theory, see [BW93]. Our algorithm also requires computing ..."
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In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Gröbner bases theory, see [BW93]. Our algorithm also requires computing computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated Kalgebras. 1
Automorphism groups of quasigalois closed arithmetic schemes. eprint arXiv:0907.0842
"... Abstract. Assume that X and Y are arithmetic schemes, i.e., integral schemes of finite types over Spec(Z). Then X is said to be quasigalois closed over Y if X has a unique conjugate over Y in some certain algebraically closed field, where the conjugate of X over Y is defined in an evident manner. N ..."
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Abstract. Assume that X and Y are arithmetic schemes, i.e., integral schemes of finite types over Spec(Z). Then X is said to be quasigalois closed over Y if X has a unique conjugate over Y in some certain algebraically closed field, where the conjugate of X over Y is defined in an evident manner. Now suppose that φ: X → Y is a surjective morphism of finite type such that X is quasigalois closed over Y. In this paper the main theorem says that the function field k (X) is canonically a Galois extension of k (Y) and the automorphism group Aut(X/Y) is isomorphic to the Galois group Gal(k(X)/k (Y)); moreover, if φ is affine and dim X = dimY holds, X is a finite Galois cover of Y by φ and Aut(X/Y) acts transitively on the fiber φ −1 (y) at each y ∈ Y. Contents
Projective planes in algebraically closed fields
 Proc. London Math. Soc
, 1991
"... We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. ..."
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We investigate the combinatorial geometry obtained from algebraic closure over a fixed subfield in an algebraically closed field. The main result classifies the subgeometries which are isomorphic to projective planes. This is applied to give new examples of algebraic characteristic sets of matroids. The main technique used, which is motivated by classical projective geometry, is that a particular configuration of four lines and six points in the geometry indicates the presence of a connected onedimensional algebraic group.
Murre, Chow motives of elliptic modular surfaces and threefolds
, 1996
"... Abstract. The main result of this paper is the proof for elliptic modular threefolds of some conjectures formulated by the secondnamed author and shown by Jannsen to be equivalent to a conjecture of Beilinson on the filtration on the Chow groups of smooth projective varieties. These conjectures are ..."
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Abstract. The main result of this paper is the proof for elliptic modular threefolds of some conjectures formulated by the secondnamed author and shown by Jannsen to be equivalent to a conjecture of Beilinson on the filtration on the Chow groups of smooth projective varieties. These conjectures are known to be true for surfaces in general, but for elliptic modular surfaces we obtain more precise results which are then used in the proof of the conjectures for elliptic modular threefolds. Let φ: E → M be the universal elliptic curve with levelN structure, whose smooth completion is an elliptic modular surface E. An elliptic modular threefold is a desingularization 2 ˜E of the fibre product E × M E. The first main result is that there exists a decomposition of the diagonal ∆ ( 2 ˜ E) modulo rational equivalence as a sum of mutually orthogonal idempotent correspondences πi which lift the Künneth components of the diagonal modulo homological equivalence. These correspondences act on the Chow groups of 2 ˜ E, and secondly we show that πi · CHj ( 2 ˜ E) = 0 for i < j or i> 2j; the implication of this is that there is a filtration on CHj ( 2 ˜E) that has j steps, as predicted by the general conjectures. The third main result is that the first step of this filtration, the kernel of π2j acting on CHj ( 2 ˜E), coincides with the kernel of the cycle class map from CHj ( 2 ˜ E) into the cohomology H2j ( 2 ˜ E); which is to say that there is a natural, geometric description for this step of the filtration. We also identify F2 CH3 ( 2 ˜ E) as the Albanese kernel. As a byproduct of our methods we also obtain some information about the Chow groups of the Chow motives for modular forms kW defined by Scholl, for k = 1 and 2, for example that CH2 ( 1W) = CH2 Alb (E), and that CH3 ( 2W) = F3 CH3 ( 2 ˜ E) lives at the deepest level of the filtration, within the Albanese kernel.
padic abelian integrals and commutative Lie groups
 J. Math. Sci
, 1996
"... The aim of this paper is to propose an “elementary ” approach to Coleman’s theory of p−adic abelian integrals [3], [5]. Our main tool is a theory of commutative p−adic Lie groups (the logarithm map); we use neither dagger analysis nor MonskyWashnitzer cohomology theory. Notice that we also treat th ..."
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The aim of this paper is to propose an “elementary ” approach to Coleman’s theory of p−adic abelian integrals [3], [5]. Our main tool is a theory of commutative p−adic Lie groups (the logarithm map); we use neither dagger analysis nor MonskyWashnitzer cohomology theory. Notice that we also treat the case of bad reduction. We discuss interrelations between p−adic abelian integrals of of the third kind and Néron pairings on abelian varieties. A preliminary version of this paper appeared as a preprint [18] in 1990. Acknowledgments. I am deeply grateful to Yu. I. Manin, D. Bertrand and A. N. Parshin for their interest in this paper. 1. Logarithm maps Let p be a prime, Qp the field of p−adic numbers, Cp the completion of its algebraic closure. Let K be a complete subfield of Cp. Clearly, K contains Qp. We will always deal with the valuation map v: K ∗ → Q normalized by the condition v(p) = 1. We will view v as a homomorphism of the
An exploration of homotopy solving in Maple
 Proc. of the Sixth Asian Symp. on Comp. Math. (ASCM 2003). Lect. Note Series on Comput. by World Sci. Publ. 10
, 2003
"... Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exa ..."
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Homotopy continuation methods find approximate solutions of a given system by a continuous deformation of the solutions of a related exactly solvable system. There has been much recent progress in the theory and implementation of such path following methods for polynomial systems. In particular, exactly solvable related systems can be given which enable the computation of all isolated roots of a given polynomial system. Extension of such methods to determine manifolds of solutions has also been recently achieved. This progress, and our own research on extending continuation methods to identifying missing constraints for systems of differential equations, motivated us to implement higher order continuation methods in the computer algebra language Maple. By higher order, we refer to the iterative scheme used to solve for the roots of the homotopy equation at each step. We provide examples for which the higher order iterative scheme achieves a speed up when compared with the standard second order scheme. We also demonstrate how existing Maple numerical ODE solvers can be used to give a predictor only continuation method for solving polynomial systems. We apply homotopy continuation to determine the missing constraints in a system of nonlinear PDE, which is to our knowledge, the first published instance of such a calculation. 1.
Two questions on mapping class groups ∗
, 2010
"... We show that central extensions of the mapping class group Mg of the closed orientable surface of genus g by Z are residually finite. Further we give rough estimates of the largest N = Ng such that homomorphisms from Mg to SU(N) have finite image. In particular, homomorphisms of Mg into SL( [ √ g + ..."
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We show that central extensions of the mapping class group Mg of the closed orientable surface of genus g by Z are residually finite. Further we give rough estimates of the largest N = Ng such that homomorphisms from Mg to SU(N) have finite image. In particular, homomorphisms of Mg into SL( [ √ g + 1], C) have finite image. Both results come from properties of quantum representations of mapping class groups.
Gröbner bases applied to finitely generated field extensions
 J. SYMBOLIC COMPUTATION
, 2000
"... Using a constructive fieldideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensions k(�x)/k(�g), resp. how to determine the (separable) degree if k(�x)/k(�g) is algebraic. Moreover, this correspondence is u ..."
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Using a constructive fieldideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensions k(�x)/k(�g), resp. how to determine the (separable) degree if k(�x)/k(�g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(�x)/k(�g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the fieldideal correspondence can also be used to determine the elements of the Galois group.