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This result is essentially the best possible, except perhaps for the values of the constants and for the exponent (n + 1)2 in the left hand side. S. Ji, J. Kollár and B. Shiffman [J-K-S] have recently proved a similar result for polynomials over a field of arbitrary characteristic without this exponent but with an ineffective dependence on the coefficients. In spite of that, we can look for other relations between the values of the fi’s and the distance to their common zeros in C. For a polynomial f ∈ Z[x1, . . . , xn] we denote its size (= degree + logarithmic height) by t(f); for α ∈ C we also denote by t(α) the minimum size of a non-zero polynomial f ∈ Z[x1, . . . , xn] for which f(α) = 0 (if there are no such polynomials we put t(α) =∞). In this paper we deal with the following problem:

Values of polynomials with integer coefficients and distance to their common zeros