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.