Bounds are obtained for sums \(\sum_x e(p^{-1}\mathfrak P(x))\) where \(e(z) =e^{2\pi iz}\), where \(\mathfrak P\) is a polynomial in \(s\) variables with coefficients in the prime field \(\mathbb F_p\), and where the sum is either over \(\mathbb F_p\), or over \(x=(x_1,\ldots, x_n)\) with \(| x_i|\leq p^{\delta}\) where \(\delta>(\deg \mathfrak P)^{-1}\). When \(\mathfrak P\) is a form of degree \(> 1\), the bounds depend on \(h=h(\mathfrak P)\), which is the least number such that \(\mathfrak P\) may be written as \(\mathfrak P=\mathfrak A_1\mathfrak B_1+\ldots+\mathfrak A_n\mathfrak B_n\) with forms \(\mathfrak A_i\), \(\mathfrak B_i\) of positive degree and with coefficients in \(\mathbb F_p\). For complete sums, i.e., sums \(S\) over \(\mathbb F_p^s\), one obtains \(| S|\ll p^{s-\kappa}\) with \(\kappa=h\rho(d)\) where \(\rho(d)>0\) depends only on the degree of \(\mathfrak P\), and the constant in \(\ll\) depends only on \(d,s\). A similar result is obtained for incomplete sums. A consequence is that a homogeneous congruence of degree \(d\) modulo \(p\) has a zero with \(| x_i| \ll p^{(1/2) +\varepsilon}\), provided the number \(s\) of variables exceeds \(s_0(d,\varepsilon)\). The proofs depend on the author’s work ”The density of integer points on homogeneous varieties” [Acta Math. 154, 243–296 (1985; Zbl 0561.10010)].