On computation of the inverse of a polynomial map over finite fields using the reduced Koopman dual linear map

This paper proposes a symbolic representation of non-linear maps $F$ in \ff^n in terms of linear combination of basis functions of a subspace of (\ff^n)^0, the dual space of \ff^n. Using this representation, it is shown that the inverse of $F$ whenever it exists can also be represented in a similar symbolic form using the same basis functions (using different coefficients). This form of representation should be of importance to solving many problems of iterations or compositions of non-linear maps using linear algebraic methods which would otherwise require solving hard computational problems due to non-linear nature of $F$.Comment: 10 page

Deformation techniques for sparse systems

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral deformation of Huber and Sturmfels. The complexity of our algorithm is quadratic in the size of the combinatorial structure of the input system. This size is mainly represented by the mixed volume of Newton polytopes of the input polynomials and an arithmetic analogue of the mixed volume associated to the deformations under consideration