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INTEGRAL POINTS OF SMALL HEIGHT OUTSIDE OF A HYPERSURFACE
, 2004
"... Let F be a nonzero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear fo ..."
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Cited by 7 (5 self)
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Let F be a nonzero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear
TOTALLY ISOTROPIC SUBSPACES OF SMALL HEIGHT IN QUADRATIC SPACES
"... Abstract. Let K be a global field or Q, F a nonzero quadratic form on K N , N ≥ 2, and V a subspace of K N . We prove the existence of an infinite collection of finite families of smallheight maximal totally isotropic subspaces of (V, F ) such that each such family spans V as a Kvector space. Thi ..."
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Abstract. Let K be a global field or Q, F a nonzero quadratic form on K N , N ≥ 2, and V a subspace of K N . We prove the existence of an infinite collection of finite families of smallheight maximal totally isotropic subspaces of (V, F ) such that each such family spans V as a Kvector space
SMALL HEIGHT IN FIELDS GENERATED BY SINGULAR MODULI
, 2015
"... We prove that some fields generated by jinvariants of CM elliptic curves (of infinite dimension over Q) satisfy the Property (B). The singular moduli are chosen so as to have supersingular reduction simultaneously above a fixed prime q, which provides strong qadic estimates leading to an explici ..."
Cache Oblivious Search Trees via Binary Trees of Small Height
"... We propose a version of cache oblivious search trees which is simpler than the previous proposal of Bender, Demaine and FarachColton and has the same complexity bounds, in particular, our data structure avoids the use of weight balanced Btrees, and can be implemented as just a single array of data ..."
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Cited by 59 (7 self)
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, and range queries in worst case O(log B n + k/B) memory transfers, where k is the size of the output. The basic idea of our data structure is to maintain a dynamic binary tree of height log nlO(1) using existing methods, embed this tree in a static binary tree, which in turn is embedded in an array in a
Reducing lattice bases to find smallheight values of univariate polynomials
 in [13] (2007). URL: http://cr.yp.to/papers.html#smallheight. Citations in this document: §A
, 2004
"... Abstract. This paper generalizes several previous results on finding divisors in residue classes (Lenstra, Konyagin, Pomerance, Coppersmith, HowgraveGraham, Nagaraj), finding divisors in intervals (Rivest, Shamir, Coppersmith, HowgraveGraham), finding modular roots (Hastad, Vallée, Girault, Toffin ..."
Algebraic points of small height missing a union of varieties
, 2008
"... Let K be a number field, Q, or the field of rational functions on a smooth projective curve of genus 0 or 1 over a perfect field, and let V be a subspace of K N, N ≥ 2. Let ZK be a union of varieties defined over K such that V � ZK. We prove the existence of a point of small height in V \ ZK, provi ..."
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Cited by 5 (5 self)
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Let K be a number field, Q, or the field of rational functions on a smooth projective curve of genus 0 or 1 over a perfect field, and let V be a subspace of K N, N ≥ 2. Let ZK be a union of varieties defined over K such that V � ZK. We prove the existence of a point of small height in V \ ZK
INTEGRAL ORTHOGONAL BASES OF SMALL HEIGHT FOR REAL POLYNOMIAL SPACES
"... Abstract. Let PN (R) be the space of all real polynomials in N variables with the usual inner product 〈 , 〉 on it, given by integrating over the unit sphere. We start by deriving an explicit combinatorial formula for the bilinear form representing this inner product on the space of coefficient vect ..."
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vectors of all polynomials in PN (R) of degree ≤ M. We exhibit two applications of this formula. First, given a finite dimensional subspace V of PN (R) defined over Q, we prove the existence of an orthogonal basis for (V, 〈 , 〉), consisting of polynomials of small height with integer coefficients
Rational Approximation To Algebraic Numbers Of Small Height: The Diophantine Equation ...
 1, J. Reine Angew. Math
"... Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multidimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we s ..."
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Cited by 28 (5 self)
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Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multidimensional "hypergeometric method" for rational and algebraic approximation to algebraic numbers. Consequently, if a; b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers x; y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pad'e approximations to systems of binomial functions, together with new Chebyshevlike estimates for primes in arithmetic progressions and a variety of computational techniques. 1.
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