Browsing Waterloo Research by Author "Liu, Yu-Ru"
Now showing items 21-31 of 31
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A prime analogue of the Erdös-Pomerance conjecture for elliptic curves
Liu, Yu-Ru (European Mathematical Society, 2005-12-31)Let E/Q be an elliptic curve of rank ≥ 1 and b ∈ E(Q) a rational point of infinite order. For a prime p of good reduction, let gb(p) be the order of the cyclic group generated by the reduction b of b modulo p. We denote ... -
Prime analogues of the Erdős–Kac theorem for elliptic curves
Liu, Yu-Ru (Elseiver, 2006-08)Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #E(Fp). We prove ... -
Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication
Liu, Yu-Ru (Wiley, 2005)Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. Denote by ω(#E(Fp)) the number of distinct prime divisors of #E(Fp). For an elliptic ... -
Roth's theorem on systems of linear forms in function fields
Liu, Yu-Ru; Spencer, Craig V.; Zhao, Xiaomei (Institute of Mathematics, 2010)1. Introduction. For r, s ∈ N = {1, 2, . . .} with s ≥ 2r + 1, let (bi,j ) be an r×s matrix whose elements are integers. Suppose that bi,1+· · ·+bi,s = 0 (1 ≤ i ≤ r). Suppose further that among the columns of the matrix, ... -
The Shifted Turan Sieve Method on Tournaments
Kuo, Wentang; Liu, Yu-Ru; Ribas, Savio; Zhou, Kevin (Cambridge University Press, 2019)Abstract. We construct a shi ed version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of ... -
The shifted Turan sieve method on tournaments II
Kuo, Wentang; Liu, Yu-Ru; Ribas, Savio; Zhou, Kevin (Elsevier, 2021-12)In a previous work [5], we developed the shifted Turán sieve method on a bipartite graph and applied it to problems on cycles in tournaments. More precisely, we obtained upper bounds for the number of tournaments which ... -
Sieve methods in combinatorics
Liu, Yu-Ru; Murty, M. Ram (Elsevier, 2005-07)We develop the Turán sieve and a ‘simple sieve’ in the context of bipartite graphs and apply them to various problems in combinatorics. More precisely, we provide applications in the cases of characters of abelian groups, ... -
Sieve Methods in Random Graph Theory
Liu, Yu-Ru; Saunders, J.C. (Springer, 2023-04-03)In this paper, we apply the Turán sieve and the simple sieve developed by R. Murty and the first author to study problems in random graph theory. In particular, we obtain upper and lower bounds on the probability of a ... -
The Unrestricted Variant of Waring's Problem in Function Fields
Liu, Yu-Ru; Wooley, Trevor D. (Adam Mickiewicz University, 2007-09)Let J k q [t] denote the additive closure of the set of k th powers in the polynomial ring Fq[t], defined over the finite field Fq having q elements. We show that when s>k + 1 and q>k 2k+2 , then every polynomial in ... -
Waring's problem in function fields
Liu, Yu-Ru; Wooley, Trevor D. (De Gruyter, 2010)Let Fq½t denote the ring of polynomials over the finite field Fq of characteristic p, and write Jk q ½t for the additive closure of the set of kth powers of polynomials in Fq½t. Define GqðkÞ to be the least integer s ... -
A weighted Turán sieve method
Liu, Yu-Ru; Murty, M. Ram (Elsevier, 2006-01)We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f (p) where p is a prime and f (x) a polynomial with integer coefficients.