Browsing University of Waterloo by Author "Liu, Yu-Ru"
Now showing items 1-20 of 31
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The asymptotic estimates and Hasse principle for multidimensional Waring's problem
Kuo, Wentang; Liu, Yu-Ru; Zhao, Xiaomei (Elsevier, 2019-09-07)Motivated by the asymptotic estimates and Hasse principle for multidimensional Waring's problem via the circle method, we prove for the first time that the corresponding singular series is bounded below by an absolute ... -
Bounds on 10th moments of (x, x^3) for ellipsephic sets
Anderson, Theresa C.; Hu, Bingyang; Liu, Yu-Ru; Talmage, Alan (University of Waterloo, 2023)Let A be an ellipsephic set which satis es digital restrictions in a given base. Using the method developed by Hughes and Wooley, we bound the number of integer solutions to the system of equations X2 i=1 x3i ... -
A Carlitz module analogue of a conjecture of Erdos and Pomerance
Kuo, Wentang; Liu, Yu-Ru (American Mathematical Society, 2009-09)Abstract. Let A = Fq[T] be the ring of polynomials over the finite field Fq and 0 = a ∈ A. Let C be the A-Carlitz module. For a monic polynomial m ∈ A, let C(A/mA) and ¯a be the reductions of C and a modulo mA respectively. ... -
Cyclicity of finite Drinfeld modules
Kuo, Wentang; Liu, Yu-Ru (Wiley, 2009-12)Le tA=Fq[T] be the polynomial ring over the finite field Fq,letk=Fq(T) be the rational function field, and let K be a finite extension of k. For a prime P of K, we denote by OP the valuation ring of P, by MP the maximal ... -
Equidistribution of Polynomial Sequences in Function Fields, with Applications
Hoang Le, Thai; Liu, Yu-Ru; Wooley, Trevor D. (University of Waterloo, 2023)We provide a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case in which the degree of the polynomial is greater than or equal to the characteristic ... -
The Erdős Theorem and the Halberstam Theorem in function fields
Liu, Yu-Ru (Institute of Mathematics of the Polish Academy of Sciences, 2004)Introduction. For n ∈ N, define ω(n) to be the number of distinct prime divisors of n. The Tur´an Theorem [9] concerns the second moment of ω(n) and it implies a result of Hardy and Ramanujan [4] that the normal order of ... -
The Erdős–Kac theorem and its generalizations
Kuo, Wentang; Liu, Yu-Ru (American Mathematical Society, 2008)Abstract. We give a survey of the Erd}o-Kac theorem and its various generalizations. In particular, we discuss an open conjecture of Erd}os and Pomerance about the distribution of the number of distinct prime divisors of ... -
Gaussian Laws on Drinfeld Modules
Kuo, Wentang; Liu, Yu-Ru (World Scientific, 2009)Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, ... -
A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression
Liu, Yu-Ru; Spencer, Craig V. (Springer, 2009-01-31)Let r1, . . . , rs be non-zero integers satisfying r1 + · · · + rs = 0. Let G Z/k1Z⊕· · ·⊕Z/knZ be a finite abelian group with ki |ki−1(2 ≤ i ≤ n), and suppose that (ri , k1) = 1(1 ≤ i ≤ s). Let Dr(G) denote the maximal ... -
A generalization of Meshulam's theorem on subsets of finite abelian groups with no 3-term arithmetic progression (II)
Liu, Yu-Ru; Spencer, Craig V.; Zhao, Xiaomei (Elsevier, 2011-02)Let G ≃ Z/k1Z ⊕ · · · ⊕ Z/kN Z be a finite abelian group with ki |ki−1 (2 ≤ i ≤ N). For a matrix Y = (ai,j) ∈ Z R×S satisfying ai,1 + · · · + ai,S = 0 (1 ≤ i ≤ R), let DY (G) denote the maximal cardinality of a set ... -
A Generalization of Roth's Theorem in Function Fields
Liu, Yu-Ru; Zhao, Xiaomei (University of Michigan, Department of Mathematics, 2012-11)For n ∈ N = {1, 2, ...}, let D3([1, n]) denote the maximal cardinality of an integer subset of [1, n] containing no nontrivial 3-term arithmetic progression. In a fundamental paper [9], Roth proved that D3([1, n]) n/log ... -
A generalization of Roth's theorem in function fields
Liu, Yu-Ru; Spencer, Craig V. (World Scientific Publishing, 2009-11)Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let formula denote the subset of 𝔽q[t] containing all polynomials of degree strictly less than N. For non-zero elements r1, …, rs of 𝔽q satisfying r1 ... -
A Generalization of the Erdös-Kac Theorem and its Applications
Liu, Yu-Ru (Cambridge University Press, 2004-12-01)We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties ... -
A Generalization of the Turán Theorem and Its Applications
Liu, Yu-Ru (Cambridge University Press, 2004-12-01)We axiomatize the main properties of the classical Turan Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties ... -
Multidimensional Vinogradov-type Estimates in Function Fields
Kuo, Wentang; Liu, Yu-Ru; Zhao, Xiaomei (Cambridge University Press, 2014)Let Fq[t] denote the polynomial ring over the finite field Fq. We employ Wooley’s new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in Fq[t]. These results allow us to apply a ... -
A note on character sums in finite fields
Bhowmick, Abhishek; Hoang Le, Thai; Liu, Yu-Ru (Elsevier, 2017-07)We prove a character sum estimate in Fq[t] and answer a question of Shparlinski. -
Number of prime factors with a given multiplicity
Elma, Ertan; Liu, Yu-Ru (Cambridge University Press, 2022-03)Let k ⩾ 1 be a natural number and ωk (n) denote the number of distinct prime factors of a natural number n with multiplicity k. We estimate the first and second moments of the functions ωk with k ⩾ 1. Moreover, we prove ... -
On sets of polynomials whose difference set contains no squares
Hoang Le, Thai; Liu, Yu-Ru (Institute of Mathematics: Polish Academy of Sciences, 2013)Let Fq[t] be the polynomial ring over the finite field Fq, and let GN be the subset of Fq[t] containing all polynomials of degree strictly less than N. Define D(N) to be the maximal cardinality of a set A⊆GN for which A−A ... -
On the number of irreducible factors with a given multiplicity in function fields
Das, Sourabhashis; Elma, Ertan; Kuo, Wentang; Liu, Yu-Ru (Elsevier, 2023-12)Let k ≥ 1 be a natural number and f ∈ Fq[t] be a monic polynomial. Let ωk(f) denote the number of distinct monic irreducible factors of f with multiplicity k. We obtain asymptotic estimates for the first and the second ... -
A Prime Analogue of Roth's Theorem in Function Fields
Liu, Yu-Ru; Spencer, Craig V. (Springer New York, 2015)Abstract. Let Fq[t] denote the polynomial ring over the nite eld Fq, and let PR denote the subset of Fq[t] containing all monic irreducible polynomials of degree R. For non-zero elements r = (r1; r2; r3) of Fq satisfying ...