The Harnack-Thom inequality for a critical point of a polynomial (Q1079619)

Let \(P\in {\mathbb{R}}[x_ 0,...,x_ n]\), 0 being an isolated critical point of P, \(P(0)=0\). For t small the Harnack-Thom inequality dim \(H_*(A_ t)\leq \dim H_*({\mathbb{C}}A_ t)\) (homologies over \({\mathbb{F}}_ 2)\) is known for the surfaces \(A_ t=\{x\in {\mathbb{R}}^{n+1}:\quad P(x)=t\},\quad {\mathbb{C}}A_ t=\{z\in {\mathbb{C}}^{n+1}:\quad P(z)=t\}.\) In the paper this inequality is sharpened: \[ \dim H_*(A_ t)\leq \dim H_*({\mathbb{C}}A_ t)^ h=\dim H_*({\mathbb{C}}A_ t)-rang M. \] Here \(H_*({\mathbb{C}}A_ t)^ h\) is the invariant subspace of the monodromy operator and M is a matrix of intersections over \({\mathbb{F}}_ 2\).