On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

On Hilbert’s solution of Waring’s problem

In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x ₁ ^k + x ₂ ^k + … + x _g ^k , where the x _i are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying, in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine how far elementary techniques of this nature can be pushed.     

Waring’s problem for integral quaternions

A (Lipschitz) integral quaternion is a Hamiltonian quaternion of the form $a+bi+cj+dk$ with all of $a,b,c,d\in \mathbb{Z}$. In 1946, Niven showed that the integral quaternions expressible as a sum of squares of integral quaternions are precisely those for which $2\mid b,c,d$; moreover, all of these are expressible as sums of three squares. Now let $m$ be an integer with $m>2$, and suppose $2^r\parallel m$. We show that if $r=0$ (i.e., $m$ is odd), then all integral quaternions are sums of $m$th powers, while if $r>0$, then the integral quaternions that are sums of $m$th powers are precisely those for which $2^r\mid b,c,d$ and $2^{r+1}\mid b+c+d$. Moreover, all of these are expressible as a sum of $g\left(m\right)$$m$th powers, where $g\left(m\right)$ is a positive integer depending only on $m$.     

Waring’s problem for special kind polynomials

An estimate of the number of summands in the generalized Waring problem for odd integervalued polynomials is given in the paper.     

Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N. As an application, we consider a Waring-type problem with sums of mixed powers.     

On algebraic relations for Ramanujan’s functions

Let P,Q, and R denote the Ramanujan Eisenstein series. We compute algebraic relations in terms of P(q ⁱ ) (i=1,2,3,4), Q(q ⁱ ) (i=1,2,3), and R(q ⁱ ) (i=1,2,3). For complex algebraic numbers q with 0<|q|<1 we prove the algebraic independence over ? of any three-element subset of {P(q),P(q ²),P(q ³),P(q ⁴)} and of any two-element subset of {Q(q),Q(q ²),Q(q ³)} and {R(q),R(q ²),R(q ³)}, respectively. For all the results we use some expressions of $P(q^{i_{1}}), Q(q^{i_{2}}) $ , and $R(q^{i_{3}}) $ in terms of theta constants. Computer-assisted computations of functional determinants and resultants are essential parts of our proofs.     

Waring’s problem with digital restrictions

The aim of this paper is to consider an analogue of Waring’s problem with digital restrictions. In particular, we prove the following result. Lets _q (n) be theq-adic sum of digits function and leth,m be fixed positive integers. Then fors>2 ^k there existsn ₀∈ℕ such that each integern≥n ₀ has a representation of the form We will even give an asymptotic formula for the number of representations ofn in this way. The result is shown with help of the circle method in combination with a “digital” version of Weyl’s Lemma. Dedicated to Professor Hillel Furstenberg The first author was supported by the Austrian Science Foundation project S8310. The second author was supported by the Austrian Science Foundation project S8308.     

Waring’s problem in prime numbers

The existence of a constant V(n) such that any sufficiently large natural number can be represented as a sum of nth degrees of primes in total quantity not exceeding this constant is proved.     

On waring’s problem for cubes

Proceedings - Mathematical Sciences -     

Residue integrals and Waring’s formulas for algebraic or even transcendental systems

The present article is focused on the study of a special class of systems of non-linear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of study of such systems we develop a method for computing residue integrals with integration over certain cycles. We describe conditions under which the mentioned residue integrals coincide with power sums of the inverses to the roots of a system of equations (i.e. multidimensional Waring’s formulas). Based on this, we develop an algorithm that computes such power sums without computing the roots. As an application of the suggested method, we consider a problem of finding sums of multi-variable number series.     

On Kulikov’s problem

Kulikov has given an étale morphism of degree d > 1 which is surjective modulo codimension two with X simply connected, settling his generalized jacobian problem. His method reduces the problem to finding a hypersurface and a subgroup of index d generated by geometric generators. By contrast we show that if D has simple normal crossings away from a set of codimension three and meets the hyperplane at infinity transversely, then necessarily d = 1. Received: 21 November 2006     

On Taylor’s formula for functions of several variables

Elementary courses in mathematical analysis often mention some trick that is used to construct the remainder of Taylor’s formula in integral form. The trick is based on the fact that, differentiating the difference $f(x) - f(t) - f'(t)\frac{{(x - t)}} {{1!}} - \cdots - f^{(r - 1)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ between the function and its degree r ? 1 Taylor polynomial at t with respect to t, we obtain $ - f^{(r)} (t)\frac{{(x - t)^{r - 1} }} {{(r - 1)!}} $ , so that all derivatives of orders below r disappear. The author observed previously a similar effect for functions of several variables. Differentiating the difference between the function and its degree r ? 1 Taylor polynomial at t with respect to its components, we are left with terms involving only order r derivatives. We apply this fact here to estimate the remainder of Taylor’s formula for functions of several variables along a rectifiable curve.     

On Dedekind’s problem for complete simple games

We state an integer linear programming formulation for the unique characterization of complete simple games, i.e. a special subclass of monotone Boolean functions. In order to apply the parametric Barvinok algorithm to obtain enumeration formulas for these discrete objects we provide a tailored decomposition of the integer programming formulation into a finite list of suitably chosen sub-cases. As for the original enumeration problem of Dedekind on Boolean functions we have to introduce some parameters to be able to derive exact formulas for small parameters. Recently, Freixas et al. have proven an enumeration formula for complete simple games with two types of voters. We will provide a shorter proof and a new enumeration formula for complete simple games with two minimal winning vectors.     

On Brauer’s problem 21

For ap-blockB of a finite groupG, we give a bound of the order of its defect groupD in terms ofk(B), the number of the irreducible ordinary characters inB.     

On Brauer’s problem 21

For ap-blockB of a finite groupG, we give a bound of the order of its defect groupD in terms ofk(B), the number of the irreducible ordinary characters inB.