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Sparsity of the intersection of polynomial images of an interval

Mei-Chu Chang — 2014

Acta Arithmetica

We show that the intersection of the images of two polynomial maps on a given interval is sparse. More precisely, we prove the following. Let f ( x ) , g ( x ) p [ x ] be polynomials of degrees d and e with d ≥ e ≥ 2. Suppose M ∈ ℤ satisfies p 1 / E ( 1 + κ / ( 1 - κ ) > M > p ε , where E = e(e+1)/2 and κ = (1/d - 1/d²) (E-1)/E + ε. Assume f(x)-g(y) is absolutely irreducible. Then | f ( [ 0 , M ] ) g ( [ 0 , M ] ) | M 1 - ε .

On sum-product representations in q

Mei-Chu Chang — 2006

Journal of the European Mathematical Society

The purpose of this paper is to investigate efficient representations of the residue classes modulo q , by performing sum and product set operations starting from a given subset A of q . We consider the case of very small sets A and composite q for which not much seemed known (nontrivial results were recently obtained when q is prime or when log | A | log q ). Roughly speaking we show that all residue classes are obtained from a k -fold sum of an r -fold product set of A , where r log q and log k log q , provided the residue sets...

Sum-product theorems and incidence geometry

Mei-Chu ChangJozsef Solymosi — 2007

Journal of the European Mathematical Society

In this paper we prove the following theorems in incidence geometry. 1. There is δ > 0 such that for any P 1 , , P 4 , and Q 1 , , Q n 2 , if there are n ( 1 + δ ) / 2 many distinct lines between P i and Q j for all i , j , then P 1 , , P 4 are collinear. If the number of the distinct lines is < c n 1 / 2 then the cross ratio of the four points is algebraic. 2. Given c > 0 , there is δ > 0 such that for any P 1 , P 2 , P 3 2 noncollinear, and Q 1 , , Q n 2 , if there are c n 1 / 2 many distinct lines between P i and Q j for all i , j , then for any P 2 { P 1 , P 2 , P 3 } , we have δ n distinct lines between P and Q j . 3. Given c > 0 , there is...

Elements of large order on varieties over prime finite fields

Mei-Chu ChangBryce KerrIgor E. ShparlinskiUmberto Zannier — 2014

Journal de Théorie des Nombres de Bordeaux

Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

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