Degree Estimates for Polynomials Constant on a Hyperplane

The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the question discussed in this paper. Estimate the degree $d$ of a polynomial in $n$ real variables, assumed to have non-negative coefficients and to be constant on a hyperplane, in terms of the number $N$ of its terms. No such estimate is possible when $n=1$. The sharp bound $d\le 2N-3$ is known when $n=2$. This paper includes two main results. The first provides a bound, not sharp for $n\ge 3$, for all $n\ge 2$. This bound implies the more easily stated bound $d\le {4(2N-3)\over 3(2n-3)}$ for $n\ge 3$. The second result is a stabilization theorem; if $n$ is sufficiently large given $d$, then the sharp bound $d \le {N-1 \over n-1}$ holds. In this situation we determine all polynomials for which the bound is sharp.