On the Leaving Coefficients of Real Many-Variable Polynomials

Abstract

For a homogeneous polynomial P in N variables, x 1, ..., x N , of degree k, the leading terms are those which contain only one variable, raised to the power k. If 0 ≤ P ≤ 1 when all variables satisfy 0 ≤ x j ≤ 1, how large can the leading coefficients be? Estimates were given by R. Aron, B. Beauzamy, and P. Enflo ( J. Approx. Theory 74 (2) (1993), 181-198); we improve these estimates in general and solve the problem completely for k = 2 and 3. Symbolic computation (MAPLE on a Digital DecStation 5000) was heavily used at two levels: first in order to get a preliminary intuition on the concepts discussed here, and second, as symbolic manipulation on polynomials, in most proofs. Numerical analysis was made on a Connection Machine CM2, using the hypercube representation obtained by B. Beauzamy, J.-L. Frot, and C. Millour (Massively parallel computations on many-variable polynomials: When seconds count, preprint).