sequences from Hilbert's 16th problem?

[Valery Liskovets]

Dear Neil,

I can partially reply to your request.

> In his 2002 survey,
>
> MR1898209 (2003c:34001) Ilyashenko, Yu. Centennial history of Hilbert's 16th problem.  Bull. Amer. Math. Soc. (N.S.)  39  (2002),  no. 3, 301--354 (electronic). (Reviewer: Lubomir Gavrilov) 34-02 (34C07 37C10 37F75)
>
> the author mentions that Petrovskii and Landis gave an incorrect proof that
> the Hilbert number H(n) is bounded above by P_3(n), "a certain polynomial
> of degree 3 [in n]".  The reference is to:
>
> MR0073004 (17,364d)
> Petrovski\u\i, I. G.; Landis, E. M.
> On the number of limit cycles of the equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are polynomials of 2nd degree. (Russian)
> Mat. Sb. N.S. 37(79) (1955), 209--250.

No, this paper contains only a proof of H(2) = 3. The proper reference for n>2 is

[PL2] Landis, E. M., Petrovskii, I. G.;
On the number of limit cycles of the equation $dy/dx=P(x,y)/Q(x,y)$, where $P$ and $Q$ are
polynomials (Russian). Mat. Sb. N.S. 43(85) (1957), 149-168. MR 19,746c.

> My question is, what is this sequence P_3(n) ?

P_3(n) = (6n^3-7n^2-11n+16)/2 for odd n and
P_3(n) = (6n^3-7n^2+n+4)/2 for even n.

Moreover, L&P point out that by Otrokov, 1954, there exist equations with  (n^2+5n-14)/2
limit cycles (hopefully this result is correct, unlike their upper bound P_3(n).)

> Later in the article Ilyashenko mentions a second Hilbert-type
> sequence E(n) (on page 305).  Same question.

I know nothing.

Valery Liskovets