The trace is the sum of the zeros of a polynomial system. In numerical algebraic geometry, trace tests are used to check for completeness of a set of solutions of a polynomial system. When coefficients of a polynomial system are treated as parameters and allowed to vary, the fundamental group of the complement of the discriminant induces permutations of the solutions of the system. The monodromy group is the group generated by these permutations. We study the monodromy group of sparse polynomial systems resulting from treating different sets of coefficients as parameters. In order to completely characterize systems under the sparse trace test, we study cases when the monodromy group is the full symmetric group. In the setting of branched covers, where monodromy is an action of the fundamental group on a fibre, we prove that restricting to a generic coordinate line preserves full symmetric monodromy.

For a finite CW complex, one can add or retract cells to get a new CW complex. The homotopy equivalence obtained in this way is called a simple homotopy equivalence. In this talk, we will talk about how to check whether a homotopy equivalence is simple by Whitehead groups and Whitehead torsions.