Birch's theorem with shifts

Abstract

A famous result due to Birch (1961) provides an asymptotic formula for the number of integer points in an expanding box at which given rational forms of the same degree simultaneously vanish, subject to a geometric condition. We present the first inequalities analogue of Birch's theorem.2010 Mathematics Subject Classification. 11D75, 11E76. Key words and phrases. Diophantine inequalities, forms in many variables, inhomogeneous polynomials.

Simultaneous equations and inequalities

“…We finish this short exposition with the paper of Chow [Cho17] which is an inequality analogue of Birch's celebrated result [Bir61]. The interested reader may look as well in the recent breakthroughs due to Myerson [Ryd18] and [Ryd19], who obtained a remarkable improvement compared to Birch's theorem for systems of quadratic and cubic integral forms.…”