Restricted Version of the Infinitesimal Hilbert 16th Problem

Upper Bounds of Topology of Complex Polynomials in Two Variables

“…In the present paper we prove the following statements used in [4]: -upper bound of the elements of the Abelian integral matrix (1.1) (see 1.7); -lower bound of its determinant ∆(t) = ∆ H,Ω (t) = det I(t), Ω = (ω 1 , . .…”

“…Upper bounds (used in [4]) of the integrals I ij (t) are stated and proved in 1.7. The lower bound of the period determinant is stated in 1.8 and proved in Section 3.…”

“…-the minimal lengths of representatives of cycles in H 1 (S t , Z) vanishing along appropriate paths from t to the critical values (Lemma 1.42 and Corollary 1.44 in 1.5, these representatives are contained in the previous bidisc); -the intersection indices of the latter cycles (Theorem 1.46 in 1.6). The above-mentioned upper bounds of locations and lengths of vanishing cycles were used in [4] (a joint paper with Yu.S.Ilyashenko), where we studied zeros of Abelian integrals of polynomial 1-forms over real ovals of real polynomials. In [4] we gave an upper bound of the number of zeros for a wide class of real polynomials of arbitrary degree n + 1 ≥ 3 and any polynomial 1-form of smaller degree.…”

“…The above-mentioned upper bounds of locations and lengths of vanishing cycles were used in [4] (a joint paper with Yu.S.Ilyashenko), where we studied zeros of Abelian integrals of polynomial 1-forms over real ovals of real polynomials. In [4] we gave an upper bound of the number of zeros for a wide class of real polynomials of arbitrary degree n + 1 ≥ 3 and any polynomial 1-form of smaller degree. This class consists of ultra-Morse polynomials H (see Definition 1.16).…”

“…The estimate of the number of zeros is done in terms of the minimal gap between two zero lines of the higher homogeneous part h and the ratio of that between two critical values of H over the diameter of the critical value set. The key object considered in the proofs in [4] is the Abelian integral matrix I(t) = (I ij (t)), I ij (t) = δ j ⊂St ω i , where (1.1) ω i , i = 1, . .…”