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On Fully Split Lacunary Polynomials in Finite Fields

Bibak K., Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 3

197--202

EN

We estimate the number of possible degree patterns of k-lacunary polynomials of degree t< p which split completely modulo p. The result is based on a combination of a bound on the number of zeros of lacunary polynomials with some graph theory arguments.

2

Visible points on modular exponential curves

Chan T. H., Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2010

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Vol. 58, no 1

17-22

EN

We obtain an asymptotic formula for the number of visible points (x,y). that is, with gcd(x, y) = 1, which lie in the box [1, U] x [1,V] and also belong to the exponential modular curves y ≡ agx (mod p). Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

3

Exponential sums with Farey fractions

Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2009

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Vol. 57, no 2

101-107

EN

For positive integers m and N, we estimate the rational exponential sums with denominator m over the reductions modulo m of elements of the set F(N) = {s/r : r, s ∈ Z, gcd(r, s) = 1, N ≥ r > s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r, m) = 1 are considered).

4

On the Euler function on differences between the coordinates of points on modular hyperbolas

Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2008

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Vol. 56, no 1

1-7

EN

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola Ηa,p(X,Y) = {(x,y) : x,y ≡ a (mod p), 1 ≤ x ≤ X, 1 ≤ y ≤ Y}. We give asymptotic formulas for the average values (x,y)∈ ... [wzór] with the Euler function φ(k) on the difference between the components of points of Ηa,p(X,Y).

5

Visible points on curves over finite fields

Shparlinski I. E., Voloch J. F.

Bulletin of the Polish Academy of Sciences. Mathematics

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2007

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Vol. 55, no 3

193-199

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.

6

Average value of the Euler function on binary palindromes

Banks W. D., Shparlinski I. E.

Bulletin of the Polish Academy of Sciences. Mathematics

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2006

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Vol. 54, no 2

95-101

EN

We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if B2l denotes the set of binary palindromes with precisely 2l binary digits, we derive an asymptotic formula for the average value of the Euler function on B2l.