Results 11  20
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40
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.
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.
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.
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.
Alterations Can Remove Singularities.
"... This paper can be used to obtain a first impression of the subject. For more detailed definitions and a discussion on come concepts used we refer to the appendix. ..."
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This paper can be used to obtain a first impression of the subject. For more detailed definitions and a discussion on come concepts used we refer to the appendix.
LOCALISATION AND COMPLETION
, 1012
"... with an addendum on the use of BrownPeterson homology in stable homotopy by ..."
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with an addendum on the use of BrownPeterson homology in stable homotopy by
Comme Appelé du Néant— As If Summoned from the Void: The Life of Alexandre
"... part of the article will appear in the next issue of the Notices. Et toute science, quand nous l’entendons non comme un instrument de pouvoir et de domination, mais comme aventure de connaissance de notre espèce à travers les âges, n’est autre chose que cette harmonie, plus ou moins vaste et plus ou ..."
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part of the article will appear in the next issue of the Notices. Et toute science, quand nous l’entendons non comme un instrument de pouvoir et de domination, mais comme aventure de connaissance de notre espèce à travers les âges, n’est autre chose que cette harmonie, plus ou moins vaste et plus ou moins riche d’une époque à l’autre, qui se déploie au cours des générations et des siècles, par le délicat contrepoint de tous les thèmes apparus tour à tour, comme appelés du néant. And every science, when we understand it not as an instrument of power and domination but as an adventure in knowledge pursued by our species across the ages, is nothing but this harmony, more or less vast, more or less rich from one epoch to another, which unfurls over the course of generations and centuries, by the delicate counterpoint of all the themes appearing in turn, as if summoned from the void. —Récoltes et Semailles, page P20 Alexandre Grothendieck is a mathematician of immense sensitivity to things mathematical, of profound perception of the intricate and elegant lines of their architecture. A couple of high points from his biography—he was a founding member of Allyn Jackson is senior writer and deputy editor of the Notices. Her email address is axj@ams.org. the Institut des Hautes Études Scientifiques (IHÉS)
ARTIN’S CONJECTURE FOR FORMS OF DEGREE 7 AND 11
"... A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greate ..."
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A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d � variables whose coefficients are
Decidable Cases of the Rational Sequence Problem
"... Determining whether or not there is a vanishing term of a linear recurrence with rational coefficients is the Rational Sequence Problem. The decidability status of this problem is open. Several formulations of the problem are presented in this paper. A decision method for determining whether or not ..."
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Determining whether or not there is a vanishing term of a linear recurrence with rational coefficients is the Rational Sequence Problem. The decidability status of this problem is open. Several formulations of the problem are presented in this paper. A decision method for determining whether or not infinitely many terms of a linear recurrence sequence vanish is given in terms of the Residue Theorem of complex variable theory, A decidable case, based on transcenedental number Theory is also discussed. The last section details an approach to the problem based on polynomial ideal theory. Keywords: Decidability, Algebraic algorithms, computational number theory 1