Visible points on curves over finite fields

• Autorzy: Shparlinski I. E. , Voloch J. F.
• Języki publikacji: EN

Abstrakty

EN

For a prime p and an absolutely irreducible modulo p polynomial ƒ(U, V) L 1\U, V] we obtain an asymptotic formula for the number of solutions to the congruence ƒ(x,y) = a (modp) in positive integers x ≤ X, y ≤ Y, with the additional condition gcd(x,y) = 1. Such solutions have a natural interpretation as solutions which are visible from the origin. These formulas are derived on average over a for a fixed prime p, and also on average over p for a fixed integer a.

Słowa kluczowe

EN

points visible from the origin; absolutely irreducible polynomial

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 3
• Strony: 193--199
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Shparlinski I. E.

autor

Voloch J. F.

• Department of Computing, Macquarie University, Sydney, NSW 2109, Australia,

Bibliografia

• [1] F. P. Boca, C. Cobeli and A. Zaharescu, Distribution of lattice points visible from the origin, Comm. Math. Phys. 213 (2000), 433-470.
• [2] E. Bombieri, On exponential sums in finite fields, Amer. J. Math. 88 (1966), 71-105
• [3] C. Cobeli and A. Zaharescu, On the distribution of the Fp-points on an affine curve in r dimensions, Acta Arith. 99 (2001), 321-329.
• [4] A. Granville, I. E. Shparlinski and A. Zaharescu, On the distribution of rationa functions along a curve over Fp and residue races, 3. Number Theory 112 (2005), 216-237.
• [5] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, The Clarendon Press, Oxford Univ. Press, New York, 1979.
• [6] M. N. Huxley and W. G. Nowak, Primitive lattice points in convex planar domains, Acta Arith. 76 (1996), 271-283.
• [7] W. G. Nowak, Primitive lattice points inside an ellipse, Czechoslovak Math. J. 55 (2005), 519-530.
• [8] I. E. Shparlinski, Primitive points on a modular hyperbola, Bull. Polish Acad. Sci. Math. 54 (2006), 193-200.
• [9] M. Vajaitu and A. Zaharescu, Distribution of values of rational maps on the Fp-points on an affine curve, Monatsh. Math. 136 (2002), 81-86.
• [10] W. G. Zhai, On primitive lattice points in planar domains, Acta Arith. 109 (2003), 1-26.
• [11] Z. Y. Zheng, The distribution of zeros of an irreducible curve over a finite field, J. Number Theory 59 (1996), 106-118.