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1 
Über das Waringsche Problem und einige VerallgemeinerungenKempner, Aubrey J. January 1912 (has links)
Thesis (doctoral)GeorgAugustUniversität zu Göttingen, 1912. / Cover title. Vita. Includes bibliographical references.

2 
Waring’s number in finite fieldsCipra, James Arthur January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / This thesis establishes bounds (primarily upper bounds) on Waring's number in finite fields. Let $p$ be a prime, $q=p^n$, $\mathbb F_q$ be the finite field in $q$ elements, $k$ be a positive integer with $k(q1)$ and $t= (q1)/k$. Let $A_k$ denote the set of $k$th powers in $\mathbb F_q$ and $A_k' = A_k \cap \mathbb F_p$. Waring's number $\gamma(k,q)$ is the smallest positive integer $s$ such that every element of $\mathbb F_q$ can be expressed as a sum of $s$ $k$th powers. For prime fields $\mathbb F_p$ we prove that for any positive integer $r$ there is a constant $C(r)$ such that $\gamma(k,p) \le C(r) k^{1/r}$ provided that $\phi(t) \ge r$. We also obtain the lower bound $\gamma(k,p) \ge \frac {(t1)}ek^{1/(t1)}t+1$ for $t$ prime. For general finite fields we establish the following upper bounds whenever $\gamma(k,q)$ exists:
$$
\gamma(k,q)\le
7.3n\left\lceil\frac{(2k)^{1/n}}{A_k^\prime1}\right\rceil\log(k),
$$
$$
\gamma(k,q)\le8n \left\lceil{\frac{(k+1)^{1/n}1}{A^\prime_k1}}\right\rceil,
$$
and
$$
\gamma(k,q)\ll n(k+1)^{\frac{\log(4)}{n\log\kp}}\log\log(k).
$$
We also establish the following versions of the Heilbronn conjectures for general finite fields. For any $\ve>0$ there is a constant, $c(\ve)$, such that if $A^\prime_k\ge4^{\frac{2}{\ve n}}$, then $\gamma(k,q)\le c(\varepsilon) k^{\varepsilon}$. Next, if $n\ge3$ and $\gamma(k,q)$ exists, then $\gamma(k,q)\le 10\sqrt{k+1}$. For $n=2$, we have $\gamma(k,p^2)\le16\sqrt{k+1}$.

3 
Waring's problem in algebraic number fieldsAlnaser, Ala' Jamil January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Todd E. Cochrane / Let $p$ be an odd prime and $\gamma(k,p^n)$ be the smallest positive integer $s$ such that every integer is a sum of $s$ $k$th powers $\pmod {p^n}$. We establish $\gamma(k,p^n) \le [k/2]+2$ and $\gamma(k,p^n) \ll \sqrt{k}$ provided that $k$ is not divisible by $(p1)/2$. Next, let
$t=(p1)/(p1,k)$, and $q$ be any positive integer. We show that if $\phi(t) \ge q$ then $\gamma(k,p^n) \le c(q) k^{1/q}$ for some constant $c(q)$. These results generalize results known for the case of prime moduli. Next we generalize these results to a number field setting. Let $F$
be a number field, $R$ it's ring of integers and $\mathcal{P}$ a prime ideal in $R$ that lies over a rational prime $p$ with ramification index $e$, degree of inertia $f$ and put $t=(p^f1)/(p1,k)$. Let $k=p^rk_1$ with $p\nmid k_1$ and $\gamma(k,\mathcal{P}^n)$ be the smallest integer
$s$ such that every algebraic integer in $F$ that can be expressed as a sum of $k$th powers $\pmod{\mathcal{P}^n}$ is expressible as a sum of $s$ $k$th powers $\pmod {\mathcal{P}^n}$. We prove for instance that when $p>e+1$ then $\gamma(k,\mathcal{P}^n) \le c(t) p^{nf/ \phi(t)}$. Moreover, if $p>e+1$ we obtain the upper bounds $\ds{\gamma(k,\mathcal{P}^n) \le 2313 \left(\frac{k}{k_1}\right)^{8.44/\log p}+\frac{1}{2}}$ if $f=2$ or $3,$ and $\ds{\gamma(k,\mathcal{P}^n)\le 129 \left(\frac{k}{k_1}\right)^{5.55/ \log p}+\frac{1}{2}}$ if $f\ge4$. We also show that if $\mathcal{P}$ does not ramify then $\ds{\gamma(k,\mathcal{P}^n) \le \frac{17}{2} \left(\frac{k}{k_1}\right)^{2.83/ \log p}+\frac{1}{2}}$ if $f\ge 2$ and $k_1\le p^{f/2}$, and $\ds{\gamma(k,\mathcal{P}^n)\le\left(\frac{f}{p^{f/21}}\right)k}$ if $f> 2$ and $k_1> p^{f/2}$.

4 
Multiplicidade de anéis 1dimensionais e uma aplicação ao problema de WaringMessias, Daniel Correia Lemos de 28 August 2015 (has links)
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Previous issue date: 20150828 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico  CNPq / Let k be an algebraically closed eld of characteristic zero and consider the polynomial
ring S = k[x1, . . . , xn] endowed with the standard grading. The Waring's problem
for a form F ∈ S of degree d asks about the least integer s ≥ 1 for which there exist
linear forms L1, . . . , Ls ∈ S satisfying F = Σs
i=1 Ldi. Such integer is called Waring rank of F. In this dissertation, we present a solution to this problem { due to CarliniCatalisanoGeramita { in the case of monomials, as an interesting application of a theorem (due to the same authors) that establishes a lower bound for the multiplicity of (standard) graded, nitely generated, reduced, 1dimensional kalgebras. / Seja k um corpo algebricamente fechado de caracter stica zero e considere o anel
de polin^omios S = k[x1, . . . , xn] munido da gradua c~ao padr~ao. O Problema de Waring
para uma forma F ∈ S de grau d questiona a respeito do menor inteiro s ≥ 1 para o
qual existem formas lineares L1, . . . , Ls ∈ S satisfazendo F = Σs
i=1 Ldi. Tal inteiro e denominado posto de Waring de F. Nesta disserta c~ao, apresentamos uma solu c~ao deste problema { devida a CarliniCatalisanoGeramita { no caso de mon^omios, como uma
interessante aplica c~ao de um teorema (dos mesmos autores) que estabelece uma cota
inferior para a multiplicidade de k algebras graduadas (padr~ao) nitamente geradas,
reduzidas e 1dimensionais.

5 
On Gaps Between Sums of Powers and Other Topics in Number Theory and CombinatoricsGhidelli, Luca 03 January 2020 (has links)
One main goal of this thesis is to show that for every K it is possible to find K consecutive natural numbers that cannot be written as sums of three nonnegative cubes. Since it is believed that approximately 10% of all natural numbers can be written in this way, this result indicates that the sums of three cubes distribute unevenly on the real line. These sums have been studied for almost a century, in relation with Waring's problem, but the existence of ``arbitrarily long gaps'' between them was not known. We will provide two proofs for this theorem. The first is relatively elementary and is based on the observation that the sums of three cubes have a positive bias towards being cubic residues modulo primes of the form p=1+3k.
Thus, our first method to find consecutive nonsums of three cubes consists in searching them among the natural numbers that are noncubic residues modulo ``many'' primes congruent to 1 modulo 3. Our second proof is more technical: it involves the computation of the SatoTate distribution of the underlying cubic Fermat variety {x^3+y^3+z^3=0}, via Jacobi sums of cubic characters and equidistribution theorems for Hecke Lfunctions of the Eisenstein quadratic number field Q(\sqrt{3}). The advantage of the second approach is that it provides a nearly optimal quantitative estimate for the size of gaps: if N is large, there are >>\sqrt{log N}/(log log N)^4 consecutive nonsums of three cubes that are less than N. According to probabilistic models, an optimal estimate would be of the order of log N / log log N.
In this thesis we also study other gap problems, e.g. between sums of four fourth powers, and we give an application to the arithmetic of cubic and biquadratic theta series. We also provide the following additional contributions to Number Theory and Combinatorics: a derivation of cubic identities from a parameterization of the pseudoautomorphisms of binary quadratic forms; a multiplicity estimate for multiprojective Chow forms, with applications to Transcendental Number Theory; a complete solution of a problem on planar graphs with everywhere positive combinatorial curvature.

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