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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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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.more » « less
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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 $$u>2\lfloor k/2\rfloor +4$$.more » « less
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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.more » « less
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Abstract 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 conditionMore 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 .more » « less
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Abstract Let {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\in\mathbb{N}}, one has G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when {k\geqslant 14}.more » « less
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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$$d=o\!\left(k^{1/4}\right)$$.more » « less
