JP Journal of Algebra, Number Theory and Applications
Volume 23, Issue 2, Pages 171 - 185
(December 2011)
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ON UNIFORM BOUNDS FOR RATIONAL POINTS ON RATIONAL CURVES AND THIN SETS
Patrick X. Rault
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Abstract: We show that, for any the number of rational points of height less than B on the image of a quadratic map from to under certain conditions, is bounded above by where the point is that the constant C is independent of the choice of the map. R is the resultant of the map and is a nonzero integer. In the special case of quadratic plane curves, we prove a bound of which improves on a result of Browning and Heath-Brown by establishing an inverse dependence on the resultant. Heath-Brown proved that for any the number of rational points of height less than B on a degree d plane curve is Browning and Heath-Brown later proved that this result holds with for quadratic curves. It is known that Heath-Brown’s theorem is sharp apart from the e, and in fact, Ellenberg and Venkatesh have proved that there is some (depending only on d) such that the point counting function for any plane curve of positive genus is Our results shed light on the open question of whether Heath-Brown’s theorem is true with |
Keywords and phrases: height, uniform, rational point, rational curve. |
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