Let K be a field of characteristic zero and K be an algebraic closure of K. Consider a sequence of polynomials G = (g 1 ,. .. , g s) in K[X 1 ,. .. , X n ], a polynomial matrix F = [f i,j ] ∈ K[X 1 ,. .. , X n ] p×q , with p ≤ q, and the algebraic set V p (F, G) of points in K at which all polynomials in G and all p-minors of F vanish. Such polynomial systems appear naturally in e.g. polynomial optimization, computational geometry. We provide bounds on the number of isolated points in V p (F, G) depending on the maxima of the degrees in rows (resp. columns) of F. Next, we design homotopy algorithms for computing those points. These algorithms take advantage of the determinantal structure of the system defining V p (F, G). In particular, the algorithms run in time that is polynomial in the bound on the number of isolated points.

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Let K be a field and φ, f = (f 1 , . . . , f s ) in K[x 1 , . . . , x n ] be multivariate polynomials (with s < n) invariant under the action of S n , the group of permutations of {1, . . . , n}. We consider the problem of computing the points at which f vanish and the Jacobian matrix associated to f , φ is rank deficient provided that this set is finite.We exploit the invariance properties of the input to split the solution space according to the orbits of S n . This allows us to design an algorithm which gives a triangular description of the solution space and which runs in time polynomial in d s , n+d d and n s+1 where d is the maximum degree of the input polynomials. When d, s are fixed, this is polynomial in n while when s is fixed and d n this yields an exponential speed-up with respect to the usual polynomial system solving algorithms.