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Rikard Bøgvad 《代数通讯》2018,46(6):2476-2487
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We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {rn(z)}n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {rn(z)}n∈N in the complement CP1?Df, where Df is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {rn(z)}n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis. 相似文献
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We prove that as n → ∞, the zeros of the polynomial cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter α and partially proves a conjecture made by the authors in an earlier work.
相似文献
$$_2 F_1 \left[ {\begin{array}{*{20}c}{ - n,\alpha n + 1} \\{\alpha n + 2} \\\end{array} ;z} \right]$$
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To each simply connected topological space is associated a graded Lie algebra; the rational homotopy Lie algebra. The Avramov-Felix
conjecture says that for a space of finite Ljusternik-Schnirelmann category this Lie algebra contains a free Lie subalgebra
on two generators. We prove the conjecture in the case when the Lie algebra has depth one. 相似文献
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Rikard Bö gvad Rolf Kä llströ m 《Transactions of the American Mathematical Society》2007,359(5):2075-2108
We study, in the setting of algebraic varieties, finite-dimensional spaces of functions that are invariant under a ring of differential operators, and give conditions under which acts irreducibly. We show how this problem, originally formulated in physics, is related to the study of principal parts bundles and Weierstrass points, including a detailed study of Taylor expansions. Under some conditions it is possible to obtain and as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of are of this type, and that there are many more--in particular, arising from toric varieties.
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A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for a line bundle on a smooth toric variety over a field of positive characteristic, the direct image under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators , and -linearized sheaves.
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