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Full-Text Articles in Physical Sciences and Mathematics
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
Siegel’S Lemma With Additional Conditions, Lenny Fukshansky
CMC Faculty Publications and Research
Let K be a number field, and let W be a subspace of K-N, N >= 1. Let V-1,..., V-M be subspaces of KN of dimension less than dimension of W. We prove the existence of a point of small height in W\boolean OR(M)(i=1) V-i, providing an explicit upper bound on the height of such a point in terms of heights of W and V-1,..., V-M. Our main tool is a counting estimate we prove for the number of points of a subspace of K-N inside of an adelic cube. As corollaries to our main result we derive an explicit …
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
Integral Points Of Small Height Outside Of A Hypersurface, Lenny Fukshansky
CMC Faculty Publications and Research
Let F be a non-zero 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 forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.