I am trying to understand a construction sketched in the paper by Gwozdziewicz, Kurdyka and Parusinski in the Proceedings of the AMS 1999 (paper here) and I'd like to request some help. The construction begins at the end of penultimate page of the text of the paper and goes on till the very end.
As can be inferred from the title, the paper is about the number of solutions of a bi-variate non-zero polynomial $P(x, y)$ of degree $d$ on the graph of $y = e^x + \sin{x}$. They prove that this is not greater than $2(d + 2)^{12}$. They then discuss a consequence of this result to o-minimal structures. In the discussion, they prove the following result.
For analytic $f: (a, \infty) \to \mathbb{R}$, let $A(d)$ denote the number of isolated solutions to the system $P(x, y) = 0, y = f(x), x > a.$
If we are given a sequence $\mathbb{N} \ni d \to a(d) \in \mathbb{N}$, then there exists an analytic function $f : (a, \infty) \to \mathbb{R}$, subanalytic at infinity, and an increasing sequence $k \to d_k$ of integers such that $$a(d_k) \le A(d_k),$$ for all $k \in \mathbb{N}$.
They sketch the idea of the construction in the paper, and it is not clear to me. I will type their construction almost verbatim here in the question, and then ask specific questions.
One can easily construct by induction: a sequence $b_k \in \mathbb{N}$, two sequences $\epsilon_k > 0$, $\eta_k > 0$, and a sequence of Polynomials $P_k = c_{1 + b_k}t^{1 + b_k} + \ldots + c_{b_{k+1}}t^{b_{k+1}}$ such that
$\|P_k\| \le \epsilon_k$,
if $r: (0, 1) \to \mathbb{R}$ is continuous, $\sup_{t \in (0, 1)} |r(t)| \le \eta_k$, then $$\#\{t \in (0, 1): P_k(t) + r(t) = 0\} \ge a(4b_k)$$,
$\sum_{k > n} \epsilon_k < \eta_n$, for all $n \in \mathbb{N}$,
where $\|\cdot\|$ is the sum of absolute value of coefficients. Now let $$g(t) = \sum_{k=1}^{\infty} P_k(t).$$ We can take $P_k$ so small that the radius of convergence of the series is $> 1$. Finally put $f(x) = g\left(\frac{x}{\sqrt{x^2 + 1}}\right), x > 0$. Let $$q_k(t, y) = y - \sum_{n=1}^{k-1} P_n(t), \qquad k > 2.$$ Clearly $q_k$ is of degree $\le b_k$ and it has at least $a(4b_k)$ zeros on the graph of g(t), for $t \in [0, 1)$. It is easy to find a polynomial $Q_k(x, y)$ of degree $\le d_k = 4b_k$ which vanishes on the zeros of $q_k\left(\frac{x}{\sqrt{x^2 + 1}, y}\right)$. Since $Q_k$ has at least $a(d_k)$ zeros on the graph of $f$, it follows that $a(d_k) \le A(d_k)$, as desired.
My questions
- Any insight on this construction would be very welcome. My current understanding is akin to hanging on to a cliff edge by my fingernails, so I'll be grateful for absolutely anything!
- Point 2 of the construction, i.e. $r(t)$ is the most unclear to me. My understanding is that they are saying that for any $r: (0, 1) \to \mathbb{R}$ that is continuous, we have that $\#\{t \in (0, 1): P_k(t) + r(t) = 0\} \ge a(4b_k)$. I am not sure how we can construct sequences that satisfy this. Absolutely no clue.
- $\|P_k\|$ is bounded from above by $\epsilon_k$, whereas $\eta_k$ is a bound on $\epsilon_{k+1} + \epsilon_{k+2} + \epsilon_{k+3} + \ldots$ according to point 3 of the construction. So when we are looking at $P_k(t) + r(t)$, $|r(t)| \le \eta_k$, and $\|P_k\| \le \epsilon_k$. I don't understand how these two interplay.
Apologies if my questions are unclear. The fact that my questions are unclear itself is evidence that my understanding of this is nebulous. I will be grateful for any help whatsoever. Thank you very much in advance.
