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Creators/Authors contains: "Wooley, Trevor D"

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  1. (, Mathematische Zeitschrift)
    Under appropriate local solubility conditions on $$\bfn$$, we obtain an asymptotic formula for $$A_{s,k}(\bfn)$$ when $$s\ge k(k+1)$$. 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. 
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    Free, publicly-accessible full text available September 1, 2026
  2. ; ; ; ; (, Indagationes Mathematicae)
    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 F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in 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. 
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    Free, publicly-accessible full text available May 1, 2026
  3. (, Springer Nature Switzerland)
    Nathanson, Melvyn B (Ed.)
    Free, publicly-accessible full text available January 1, 2026
  4. ; (, International Mathematics Research Notices)
    We present estimates for smooth Weyl sums of use on sets of major arcs in applications of the Hardy–Littlewood method. In particular, we derive mean value estimates on major arcs for smooth Weyl sums of degree $$k$$ delivering essentially optimal bounds for moments of order $$u$$ whenever $$u>2\lfloor k/2\rfloor +4$$. 
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    Free, publicly-accessible full text available November 26, 2025
  5. ; (, Journal of the European Mathematical Society)
    Let $$k$$ be a natural number and let $$c=2.134693\ldots$$ be the unique real solution of the equation $$2c=2+\log (5c-1)$$ in $$[1,\infty)$$. Then, when $$s\ge ck+4$$, 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. 
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    Free, publicly-accessible full text available November 28, 2025
  6. ; (, Functiones et Approximatio Commentarii Mathematici)
    Full Text Available 
  7. ; (, Bulletin of the London Mathematical Society)
    We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of . In particular, when and is defined via the relation , then for all large numbers there is an integer with for which . 
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    Free, publicly-accessible full text available March 1, 2026
  8. ; (, Journal für die reine und angewandte Mathematik (Crelles Journal))
    Abstract Let G ⁢ ( k ) {G(k)}denote the least numbershaving the property that everysufficiently large natural number is the sum of at mostspositive integralk-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}. 
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  9. (, Mathematical Proceedings of the Cambridge Philosophical Society)
    Abstract When$$k\geqslant 4$$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$$d=o\!\left(k^{1/4}\right)$$. 
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  10. (, Bulletin of the Australian Mathematical Society)
    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$$1\le k_1<\cdots . Subject to the condition$$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)$$. 
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