Effective Bezout Identities in Q[z1,...,Zn]
MARYLAND UNIV COLLEGE PARK DEPT OF MATHEMATICS
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If psub 1..... , Psub mare n-variate polynomials with integral coefficients and no common zeros in Cexp n, Brownawell has shown in 1986 that there exist qsub 1 ...., qsub m polynomials with integral coefficients and nu is an element of Z such that psub 1 qsub 1 ... psub m qsub m nu, and max deg qsub j max deg psub j exp n. On the other hand if h logarithm of the largest coefficient of all the psub j, and hsub 1 is the corresponding quantity for the qsub j, then there is no sharp estimate of hsub 1 in terms of h and max deg psub j. In this paper we show that when the variety of common zeros at infinity of the psub j is discrete then essentially we have hsub 1 Dexp cnh for an absolute constant c. If there were an algorithm to compute the qsub j in Dexp cn time one would obtain exactly the above estimate. Current algorithms require about Dexp n squared operations.
- Numerical Mathematics