Density of zeros for systems of forms

Source abstract

Abstract Let be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial zero. For example, may be a totally imaginary number field or a finite extension of . Suppose are a collection of forms (i.e. homogeneous polynomials) of degree over . Bik, Draisma and Snowden recently proved that there exists a constant such that the rational zeros of the collection are Zariski dense, as long as its Birch rank is greater than . We establish an effective bound for this constant, improving vastly on the astronomical bound coming from their proof. Our result has applications for surjectivity of polynomial maps and for the Hardy–Littlewood circle method.