Simultaneous integer values of pairs of quadratic forms

作者：Heath Brown D R^*; Pierce L B

来源：Journal fur die Reine und Angewandte Mathematik, 2017, 727(727): 85-143.

DOI：10.1515/crelle-2014-0112

We prove that a pair of integral quadratic forms in five or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

• 出版日期2017-6