Resultant of an equivariant polynomial system with respect to the symmetric group

The analysis and solving of polynomial systems are fundamental problems in computational algebra. In many applications, polynomial systems are highly structured and it is very useful to develop specific methods in order to take into account a particular structure. In this talk, we will focus on systems of n homogeneous polynomials f1, . . . , fn in n variables x1, . . . ,xn that are globally invariant under the action of the symmetric group Sn of n symbols. More precisely, we will assume that for any integer i ∈ {1,2, . . . ,n} and any permutation σ ∈ Sn σ( fi) := fi(xσ(1),xσ(2), . . . ,xσ(n)) = fσ(i)(x1,x2, . . . ,xn). (1)