Symbolic Computation Group

Let V be an equidimensional algebraic set defined over a field K. The \emph{Chow form} of V is a polynomial that characterizes V completely, providing in particular a parametrization or \emph{geometric resultion} of V, from which we can in turn deduce a set of defining equations. In this talk, we will discuss the following problem : given a system of polynomial equations f_1=\dots=f_s‎ = 0 with integer coefficients, defining an algebraic set with equidimensional components V_0,\dots,V_{n-1}, and respective Chow forms C_0,\dots,C_{n-1}, for which primes p does it hold that the Chow forms of the equidimensional components of the reduced system over \FF_p, equal the reductions mod p of C_0,\dots,C_{n-1} ? Such primes are deemed lucky. Using the connection between resultants and Chow forms, we give a sufficient condition for a prime p to be lucky, from which can deduce a bound on the product of bad primes, taking into account the sparsity of the system. Joint work in progress with J.Eliott, E.Schost and E.Tsigaridas.