Timeline for Distribution of number of integer solutions in box to bivariate polynomials?

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Nov 3, 2020 at 21:42	comment	added	Turbo		I think in your example d is max degree not sum and nevertheless it is helpful.
Nov 3, 2020 at 20:54	comment	added	Stanley Yao Xiao		Let $f(x,y)$ be your bivariate polynomial. By a result of Bombieri and Pila from 1989, one has that for any $\epsilon > 0$ there exists a number $c_{d,\epsilon}$ depending on $d = d_x + d_y$ and $\epsilon$ but not on $b, t$ such that the number of solutions to $f(x,y) = 0$ in $[-t,t]^2$ is at most $c_{d,\epsilon} t^{1/d + \epsilon}$. This is essentially sharp, as one can almost reach this bound by taking something like $f(x,y) = y - x^d$.
Nov 3, 2020 at 20:49	history	edited	Turbo	CC BY-SA 4.0	added 63 characters in body
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Nov 3, 2020 at 8:31	history	edited	Turbo		edited tags
Nov 3, 2020 at 8:00	history	edited	Turbo	CC BY-SA 4.0	added 58 characters in body
Nov 3, 2020 at 7:52	history	edited	Turbo	CC BY-SA 4.0	added 148 characters in body; edited tags; edited title
Nov 3, 2020 at 7:26	history	asked	Turbo	CC BY-SA 4.0