Density of solutions for systems of forms

Let K be a field of characteristic zero over which every diagonal form in sufficiently many variables admits a nontrivial solution. For example, K may be a totally imaginary number field or a finite extension of a p-adic field. Suppose f₁,…,fₛ are forms of degree d over $K.$ Bik, Draisma and Snowden recently proved that there exists a constant $B = B(d,s,K)$ such that the rational solutions to the system of equations f₁=…=fₛ = 0 are Zariski dense, as long as the Birch rank of f₁,…,fₛ is greater than $B.$ 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.