Let \(G\) be a finite group, \(V\) a nontrivial, absolutely irreducible representation of \(G\) and \(f_{\lambda}:V\to V\) a family of equivariant maps parametrized by \(\lambda\in \mathbb{R}\). One goal of (static) equivariant bifurcation theory is to obtain a classification of the possible branching patterns for generic families \(f\). One wants to know the number of subcritical and supercritical solution branches, their isotropy, stability, index. The authors study these questions for polynomial maps of the form \(f_ \lambda(x)=\lambda x+R(x)+Q(x)+o(\| x\|^ d)\) where \(R(x)=h(x)x\) is radial of degree less than \(d\) and \(Q\) is homogeneous of degree \(d\). Here \(d=d(G,V)>1\) is the smallest integer such that there exists a nonradial, equivariant, homogeneous polynomial map \(V\to V\) of degree \(d\). The main results show that the branching pattern is closely related to the zeros of the vector field \({\mathcal P}_ Q\) induced by \(Q\) on the unit sphere of \(V\). For instance, if all zeros of \({\mathcal P}_ Q\) are simple then the branching pattern is determined by \({\mathcal P}_ Q\). By a result of the first author [J. Dyn. Differ. Equations 1, No. 4, 369-421 (1989; Zbl 0693.58014)] there exists an integer \(\delta=\delta(G,V)\geq d(G,V)\) such that the bifurcation pattern for generic \(f\) is determined by the \(\delta\)-jet of \(f_ 0\) at \(0\in V\). The results of this paper are especially useful if \(\delta(G,V)=d(G,V)\). Examples and applications of the theory developed here (in particular to the existence of solution branches with submaximal isotropy group) are deferred to part II [Arch. Ration. Mech. Anal. 120, No. 2, 147-190 (1992)].