The existential Hilbert sixteenth problem: a uniform bound on the number of limit cycles of planar polynomial vector fields

Theorem 1.1: for each $d\ge1$ there is a finite integer $B(d)$ such that every real planar polynomial vector field of degree at most $d$ has at most $B(d)$ limit cycles in the whole plane, with no restriction on coefficients, location, stability or hyperbolicity. The proof works with matching systems of local passages and shows that each fixed auxiliary system has finitely many isolated solutions, uniformly as coefficients vary. It does NOT give an explicit or effective $B(d)$, so it says nothing about the value of $H(2)$, about Smale's question whether $H(d)\le Kd^q$, or about the relative positions of cycles asked for in Hilbert's full second part. The exact quintic Lienard bound in the same family is a separate entry.

Theorem 1.1: for each $d\ge1$ there is a finite integer $B(d)$ such that every real planar polynomial vector field of degree at most $d$ has at most $B(d)$ limit cycles in the whole plane, with no restriction on coefficients, location, stability or hyperbolicity. The proof works with matching systems of local passages and shows that each fixed auxiliary system has finitely many isolated solutions, uniformly as coefficients vary. It does NOT give an explicit or effective $B(d)$, so it says nothing about the value of $H(2)$, about Smale's question whether $H(d)\le Kd^q$, or about the relative positions of cycles asked for in Hilbert's full second part. The exact quintic Lienard bound in the same family is a separate entry.