Let F be the family of all polynomial vector fields of degree d in the plane. Hilbert's 16th problem conjectures that there is a finite bound on the number of limit cycles of the vector fields belonging to F. This as yet open problem (if d is at least 2) has a tantalizingly model-theoretic flavor, but no model-theoretic framework has been discovered so far to capture it. On the other hand, Roussarie's finite cyclicity conjecture reduces the problem to a localized (in the parameter space) one. In recent joint work with Kaiser and Rolin, we used o-minimality (a branch of model theory) to establish Roussarie's conjecture in a very special case. I will survey our approach, with an emphasis on the role o-minimality plays in obtaining a finite upper bound on the number of limit cycles.

We wish to compute with polynomials where the exponents are not known in advance. Expressions of this sort arise frequently in practice, for example in the analysis of algorithms, and it is difficult to work with them effectively in current computer algebra systems. There are various simple operations we must be able to perform, such as squaring $x^{2n}-1$ to get $x^{4n}-2x^{2n}+1$, or differentiating to get $2nx^{2n-1}$. Other operations are less obvious.

We consider the case where multivariate polynomials can have exponents that are themselves integer-valued multivariate polynomials. We call these objects "symbolic polynomials" and show they form a unique factorization domain, naturally related to the polynomial ring. We present algorithms to compute their GCD, factorization and functional decomposition.

This is the second talk in the Steenrod algebra learning seminar.