[For the entire collection see Zbl 0583.00005.]
Let \(F_ 1,...,F_ r\) be integral forms in \({\mathfrak x}=(x_ 1,...,x_ n)\). In this paper one is interested in bounds for the smallest non- trivial simultaneous solution of the congruences \(F_ i({\mathfrak x})\equiv 0 (mod p).\) Suppose that the \(F_ i\) all have degree d, and that the integral combinations \(\sum a_ i F_ i({\mathfrak x})\) are singular mod p only when all the \(a_ i\equiv 0 (mod p)\). Then (Theorem 1) one can find a solution \(| {\mathfrak x}| \ll p^{\rho +1/d},\) providing that \(\rho >0\) and \(n>c_ 1(d)r \rho^{-1}.\)
If one drops the non-singularity condition, and allows the forms to have degrees \(\leq d\), one still obtains (Theorem 2) a bound \(| {\mathfrak x}| \ll p^{\rho +}\) for \(n>c_ 2(d) r^{3F_{d-1}} \rho^{-2},\) where \(F_ k\) is the k-th Fibonacci number. If d and the degrees of the \(F_ i\) are all odd one can even (Theorem 3) obtain \(| {\mathfrak x}| \ll p^{\rho +1/3},\) providing that \(n>c_ 3(d) r^{6F_{d- 3}} \rho^{-2}.\) The introduction describes a number of other related results.
The proofs are based on an exponential sum estimate from the author’s work [Acta Arith. 44, 281-297 (1984; Zbl 0516.10029)], together with techniques concerning the composition of polynomials.