Density of Rational Points on a Certain Smooth Bihomogeneous Threefold
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We establish sharp upper and lower bounds for the number of rational points of bounded anticanonical height on a smooth bihomogeneous threefold defined over Q and of bidegree (1, 2). These bounds are in agreement with Manin's conjecture.
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In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater than or equal to every element of S. Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for n > 2, the Fermat curve of equation has no other rational points than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1).
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to a projective space of some dimension over K. This means that its function field is isomorphic to the field of all rational functions for some set of indeterminates, where d is the dimension of the variety. Let V be an affine algebraic variety of dimension d defined by a prime ideal I = ⟨f1, ..., fk⟩ in . If V is rational, then there are n + 1 polynomials g0, ..., gn in such that In order words, we have a of the variety.
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