Pure Maths Colloquium: Real and rational systems of forms
31 October 2016 16:00 in CM221
Consider a system f consisting of R forms of degree d with integral coefficients. We seek to estimate the number of solutions to f=0 in integers of size B or less. A classic result of Birch (1962) answers this question when the number of variables is of size at least C(d)*R^2 for some constant C(d), and the zero set f = 0 is smooth.
We reduce the number of variables needed to C'(d)*R, and give an extension to systems of Diophantine inequalities |f| < 1 with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality. We relate these results to Manin's conjecture in arithmetic geometry and to Diophantine approximation on manifolds.
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