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The number of integral points on arcs and ovals. (English) Zbl 0718.11048
Let $$C$$ be an arc of a convex curve $$y=F(x)$$ lying in the unit square, and let $$R(N)$$ be the number of integer points on $$NC$$ (rational points $$(m/N,n/N)$$ on $$C$$). The authors prove results of the form $R(N) < B(\varepsilon) N^{\theta +\varepsilon} \tag{1}$ for $$N$$ sufficiently large. V. Jarník [Math. Z. 24, 500–518 (1925; JFM 51.0153.01)] constructed curves with $$R(N)\geq AN^{2/3}$$ for any given $$N$$, so the conditions “$$N$$ sufficiently large” and “$$B(\varepsilon)$$ depending on $$C$$” cannot both be dropped. H. P. F. Swinnerton-Dyer [J. Number Theory 6, 128–135 (1974; Zbl 0285.10020)] gave (1) with $$\theta =3/5$$ when $$F(x)$$ has three continuous derivatives. W. M. Schmidt [Monatsh. Math. 99, 45–72 (1985; Zbl 0551.10026)] made $$B(\varepsilon)$$ independent of $$C$$ for $$F^{(3)}$$ non-vanishing.
Algebraic curves parametrised by polynomials of degree $$d$$ over $${\mathbb Q}$$ have $$R(N)\geq AN^{1/d}$$ for infinitely many $$N$$. The authors show (Theorems 1 and 2) that all other real-analytic curves satisfy (1) with $$\theta =0$$. The constant $$B(\varepsilon)$$ depends on $$C$$ ineffectively. For irreducible algebraic curves of degree $$d$$, (1) holds with $$\theta =1/d$$ and $$B(\varepsilon)$$ depending only on $$d$$ (Theorem 5). For an arbitrary curve (1) holds with $$\theta =1/2+8/3(d+3),$$ provided that $$F(x)$$ has $$D=(d+1)(d+2)/2$$ continuous derivatives, with $$B(\varepsilon)$$ depending on upper bounds for the derivatives of $$F(x)$$ (Theorem 6). Theorem 8 is a slightly weaker result in which $$B(\varepsilon)$$ depends on the number of zeros of $$F^{(D)}(x)$$. For $$D\geq 325$$ Swinnerton-Dyer’s exponent is greatly improved. The authors show that $$\varepsilon$$ is needed in the exponent by constructing an infinitely differentiable $$F(x)$$ for any given value of $$N$$, with $$F^{(3)}(x)$$ non-negative, but $$R(N)\geq A(N \log N)^{1/2}.$$
The method is to show that if $$D$$ integer points lie close on a smooth curve, then a certain determinant of monomials in the coordinates is small. Being an integer, the determinant must be zero, and the $$D$$ points lie on an algebraic curve of degree $$d$$ (for algebraic $$C$$ the method is modified to ensure that $$C$$ is not a component). The effective results use induction on the degree and intersection number arguments.

##### MSC:
 11P21 Lattice points in specified regions 11D99 Diophantine equations 11H06 Lattices and convex bodies (number-theoretic aspects) 11G99 Arithmetic algebraic geometry (Diophantine geometry) 14G05 Rational points 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
##### Keywords:
arcs; ovals; convex curve; number of integer points; algebraic curves
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##### References:
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