We prove 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 of the field, which is a natural barrier when applying the Weyl differencing process to function fields. We also discuss applications to van der Corput, intersective and Glasner sets in function fields.

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We show that no $2$-superirreducible polynomials exist in $\mathbb F[t]$ when $p=2$ and that no such polynomials of odd degree exist when $p$ is odd. We address the remaining case in which $p$ is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree $d$. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

Consider a set of integers 𝒜 having finite diameter X, and a system of simultaneous polynomial equations to be solved over 𝒜 . In many circumstances, it is known that the number of solutions of this system is O(X^ε |𝒜|^θ) for a suitable θ>0 and any ε >0 . These estimates become worse than trivial when the diameter X is very large compared to |𝒜| , or equivalently, when the set 𝒜 is very sparse. This motivates the problem of seeking a new set of integers ℬ , in a certain sense isomorphic to 𝒜 , having the property that the diameter X' of ℬ is smaller than X, and at the same time the set ℬ preserves the salient features of the solution set of the system of equations in question. We report on our speculative investigations concerning this problem closely associated with the topic of Freiman homomorphisms.

When s⩾ k⩾ 3 and n_1,… ,n_k are large natural numbers, denote by A_s,k(n) the number of solutions in non-negative integers x to the system x_1^j+⋯ +x_s^j=n_j (1⩽ j⩽ k). Under appropriate local solubility conditions on n , we obtain an asymptotic formula for A_s,k(n) when s⩾ k(k+1) . This establishes a local–global principle in the Hilbert–Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of alpha nk$\alpha n<^>k$. In particular, when k >= 6$k\geqslant 6$ and rho(k)$\rho (k)$ is defined via the relation rho(k)-1=k(logk+8.02113)$\rho (k)<^>{-1}=k(\log k+8.02113)$, then for all large numbers N$N$ there is an integer n$n$ with 1 <= n <= N$1\leqslant n\leqslant N$ for which parallel to alpha nk parallel to <= N-rho(k)$\Vert \alpha n<^>k\Vert \leqslant N<^>{-\rho (k)}$.

Fix k , s , n ∈ N k,s,n\in \mathbb {N} , and consider non-zero integers c 1 , … , c s c_1,\ldots ,c_s , not all of the same sign. Provided that s ⩾ k ( k + 1 ) s\geqslant k(k+1) , we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine diagonal hypersurface defined by the equation c 1 x 1 k + … + c s x s k = n c_1x_1^k+\ldots +c_sx_s^k=n . This conclusion surmounts the conventional convexity barrier tantamount to the square-root cancellation limit for this problem.

Let k be a natural number and let c = 2.134693 ... be the unique real solution of the equation 2c = 2 + log(5c- 1) in [1; infinity). For s >= ck + 4, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and s positive integral k-th powers.

Let satisfy . Freĭman's theorem shows that when , there exists such that all large integers are represented in the form , with , if and only if diverges. We make this theorem effective by showing that, for each fixed , it suffices to impose the condition More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when and , all large integers are represented in the form , with .

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weylsums. We focus attention on the problem of representing large positive integers as sums of a square and a number of k-th powers. We show that such representations exist when the number of k-th powers is at least [c(0)k]+ 2, where c0= 2.13629. . .. By developing an abstract framework capable of handling sequences with appropriate distribution properties, analogous conclusions are obtained, for example, when the square is restricted to have prime argument.

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ , we show that there is a paucity of nondiagonal solutions to the Diophantine system $\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$ .

Let $G(k)$ denote the least number $s$ such that every sufficiently large natural number is the sum of at most $s$ positive integral $k$th powers. We show that $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$, $G(12)\le 72$, $G(13)\le 81$, $G(14)\le 90$, $G(15)\le 99$, $G(16)\le 108$.

For every finite abelian group G, there are positive integers n and d such that G is isomorphic to the multiplicative group of d-th powers of reduced residues modulo n.

Abstract Let G ⁢ ( k ) {G(k)} denote the least number s having the property that every sufficiently large natural number is the sum of at most s positive integral k-th powers. Then for all k ∈ ℕ {k\in\mathbb{N}} , one has G ⁢ ( k ) ⩽ ⌈ k ⁢ ( log ⁡ k + 4.20032 ) ⌉ . G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when k ⩾ 14 {k\geqslant 14} .

We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables.

Suppose that θ is irrational. Then almost all elements ν∈Z[θ] that may be written as a k-fold product of the shifted integers n+θ (n∈N) are thus represented essentially uniquely.

AbstractWhen$k\geqslant 4$and$0\leqslant d\leqslant (k-2)/4$, we consider the system of Diophantine equations\begin{align*}x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\leqslant j\leqslant k,\, j\ne k-d).\end{align*}We show that in this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when$d=o\!\left(k^{1/4}\right)$.

We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.

When k and s are natural numbers and h is an element of Z(k), denote by J(s,k) (X; h) the number of integral solutions of the system Sigma(s)(i=1)(x(i)(j)-y(i)(j)) = h(j) (1 <= j <= k), with 1 <= x(i), y(i) <= X. When s < k(k + 1)/2 and (h(1), ..., h(k-1)) not equal 0, Brandes and Hughes have shown that J(s,k) (X; h) = o(X-s). In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for J(s,k) (X; h) in the critical case s = k(k + 1)/2. The latter requires minor arc estimates going beyond square-root cancellation.

Defining the truncated extension operator E for a sequence a(n) with n ∈ℤ by putting Ea(α,β):=∑_|n|≤ Na(n) e(α n^3 + β n), we obtain the conjectured tenth moment estimate Ea_L^10(𝕋^2)≲_ϵ N^1/10+ϵa_ℓ^2(ℤ). We obtain related conclusions when the curve (x,x^3) is replaced by (ϕ_1(x), ϕ_2(x)) for suitably independent polynomials ϕ_1(x),ϕ_2(x) having integer coefficients.