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Thursday, December 7, 2017

**Abstract:** 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.