Systems of cubic forms in many variables

Abstract

We consider systems F (x) of R homogeneous forms of the same degree d in n variables with integral coefficients. If n ≥ d2 d R + R and the coefficients of F lie in an explicit Zariski open set, we give a nonsingular Hasse principle for the equation F (x) = 0, together with an asymptotic formula for the number of solutions to in integers of bounded height. This improves on the number of variables needed in previous results for general systems F as soon as the number of equations R is at least 2 and the degree d is at least 4.

Quadratic forms and systems of forms in many variables

“…When d = 2 or 3, previous work of the author provides the same conclusion with the condition F ∈ U d,n,R (Q) replaced by the condition that V (F ) be smooth of dimension n − R − 1. See Theorem 1.2 and the comments after Lemma 1.1 in [5] for the case d = 2 and see Theorem 1.2 in [6] for the case d = 3. The case of interest in the theorem is thus d ≥ 4.…”

“…We give a proof after stating the following simple lemma on real matrices, which is Lemma 3.2(iii) in [6].…”

On the Hasse principle for complete intersections

“…There have been two recent notable breakthroughs. Myerson [16], [17] improved the square dependence on R in Birch's result to a linear one. When d = 2 and 3, these results improve the lower bound to n − σ ≥ 8R and 25R respectively.…”

Quantitative results on Diophantine equations in many variables

“…Recent results by Myerson improve on Birch' result for systems of forms when V is a complete intersection (which is implied by (1) in Birch' theorem). He shows that under this condition one can replace condition (1) by n ≥ 9R respectively n ≥ 25R for systems of degree 2 respectively 3 [Mye15,Mye17].…”