On the monodromy action for $f(x,y)=g(x)+h(y)$

Abstract

We solve the problem of determining under which conditions the monodromy of a vanishing cycle generates the whole homology of a regular fiber for a polynomial $f(x,y)=g(x)+h(y)$ where $h$ and $g$ are polynomials with real coefficients having real critical points and satisfying $deg(g)\cdot deg(h)\leq 2500$ and $\gcd(deg(g),deg(h))\leq 2$. As an application, under the same assumptions we solve the tangential-center problem which is known to be linked to the weak 16th Hilbert problem.