Yu-Ru Liu

WARING'S PROBLEM IN FUNCTION FIELDS

Abstract Let denote the ring of polynomials over the finite field 𝕗 q of characteristic p, and write for the additive closure of the set of kth powers of polynomials in . Define Gq (k) to be the least integer s satisfying the property that every polynomial in of sufficiently large degree admits a strict representation as a sum of s kth powers. We employ a version of the Hardy-Littlewood method involving the use of smooth polynomials in order to establish a bound of the shape Gq (k) ≦ Ck log k + O(k log log k). Here, the coefficient C is equal to 1 when k < p, and C is given explicitly in terms of k and p when k > p, but in any case satisfies C ≦ 4/3. There are associated conclusions for the solubility of diagonal equations over , and for exceptional set estimates in Warings problem.

Multidimensional Vinogradov-type Estimates in Function Fields

Abstract. Let Fq[t] be the ring of polynomials over the finite field Fq. In this talk, we will employ Wooley’s new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in Fq[t]. These results allow us to apply a variant of the circle method to obtain asymptotic formulas for a system connected to the problem about linear spaces lying on hypersurfaces defined over Fq[t]. This is a joint work with Wentang Kuo and Xiaomei Zhao.

A Carlitz module analogue of a conjecture of Erdos and Pomerance

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 α be the reductions of C and a modulo mA respectively. Let f α (m) be the monic generator of the ideal {f ∈ A,Cf(a) = 0} on C(A/mA). We denote by ω(f a (m)) the number of distinct monic irreducible factors of f α (m). If q ≠ 2 or q = 2 and α ≠ 1,T, or (1 + T), we prove that there exists a normal distribution for the quantity ω(f α (m))- 1/2(log deg m) 2 1/3(log deg m) 3/2 This result is analogous to an open conjecture of Erdos and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of b modulo n, where b is an integer with |b| > 1, and n a positive integer.

GAUSSIAN LAWS ON DRINFELD MODULES

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, write , where is the valuation ring of 𝔓 and its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓 acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with , we prove that there exists a normal distribution for the quantity For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓 acting on a Tate module of ϕ and obtain similar results.

A GENERALIZATION OF ROTH'S THEOREM IN FUNCTION FIELDS

Let 𝔽q[t] denote the polynomial ring over the finite field 𝔽q, and let 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 + ⋯ + rs = 0, let denote the maximal cardinality of a set which contains no non-trivial solution of r1x1 + ⋯ + rsxs = 0 with xi ∈ A (1 ≤ i ≤ s). We prove that .

A note on character sums in finite fields

Abstract We prove a character sum estimate in F q [ t ] and answer a question of Shparlinski.

A Prime Analogue of Roth’s Theorem in Function Fields

Let \(\mathbb{F}_{q}[t]\) denote the polynomial ring over the finite field \(\mathbb{F}_{q}\), and let \(\mathcal{P}_{R}\) denote the subset of \(\mathbb{F}_{q}[t]\) containing all monic irreducible polynomials of degree R. For non-zero elements r = (r1, r2, r3) of \(\mathbb{F}_{q}\) satisfying r1 + r2 + r3 = 0, let \(D(\mathcal{P}_{R}) = D_{\mathbf{r}}(\mathcal{P}_{R})\) denote the maximal cardinality of a set \(A_{R} \subseteq \mathcal{P}_{R}\) which contains no non-trivial solution of \(r_{1}x_{1} + r_{2}x_{2} + r_{3}x_{3} = 0\) with x i ∈ A R (1 ≤ i ≤ 3). By applying the polynomial Hardy-Littlewood circle method, we prove that \(D(\mathcal{P}_{R}) \ll _{q}\vert \mathcal{P}_{R}\vert /(\log \log \log \log \vert \mathcal{P}_{R}\vert )\).