Tag - Waring’s problem

Trevor Wooley: Waring’s Problem

30th November, 2022

In 1770, E. Waring made an assertion these days interpreted as conjecturing that when k is a natural number, all positive integers may be written as the sum of a number g(k) of positive integral kth powers, with g(k) finite. Since the work of Hardy and Littlewood a century ago, attention has largely shifted to the problem of bounding G(k), the least number s having the property that all sufficiently large integers can be written as the sum of s positive integral kth powers. It is known that G(2) = 4 (Lagrange), G(3) ≤ 7 (Linnik), G(4) = 16 (Davenport), and G(5) ≤ 17, G(6) ≤ 24, ..., G(20) ≤ 142 (Vaughan and Wooley). For large k one has G(k) ≤ k(log k+log log k+2+o(1)) (Wooley). We report on very recent progress joint with Joerg Bruedern. One or two new world records will be on display.

Matej Brešar: Images of non-commutative polynomials

23rd July, 2020

Let f = f(X1, ..., Xm) be a non-commutative polynomial with coefficients in a field F. We will discuss various questions concerning the image of f in an F-algebra A, which is defined to be the set f(A) = {f(a1, ..., am) | a1, ..., am ∈ A}. A special emphasis will be on the Waring type problem, asking about the existence of a positive integer N (independent of f, provided that f is neither an identity nor a central polynomial of A) such that every element, or at least every commutator, in A is a linear combination of N elements from f(A). We are primarily interested in the case where A = Mn(F), but some other algebras will also be considered.