Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
Waring problem as an issue of polynomial computer algebra. / Vavilov, Nikolai .
International Conference Polynomial Computer Algebra 2020: St Petersburg October 2020. ed. / Николай Васильев. Международный математический институт им. Эйлера, 2020.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
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TY - GEN
T1 - Waring problem as an issue of polynomial computer algebra
AU - Vavilov, Nikolai
PY - 2020/10/4
Y1 - 2020/10/4
N2 - In its original XVIII century form the classical Waring problem consisted in finding for each natural $k$ the smallest such $s=g(k)$ that all natural numbers $n$ can be written as sums of $s$ non-negative $k$-th powers, $n=x_1^k+\ldots+x_s^k$. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that {\it almost all\/} $n$ can be expressed in this form. In the XX century this problem was further specified, as for finding such $G(k)$ {\it and\/} the precise list of exceptions. In the present talk I sketch the key steps in the solution of this problem, with a special emphasis on algebraic and computational aspects. I describe various connections of this problem, and its modifications, such as the rational Waring problem, the easier Waring problem, etc., with the current research in polynomial computer algebra,especially with identities, symbolic polynomials, etc. and promote several outstanding computational challenges.
AB - In its original XVIII century form the classical Waring problem consisted in finding for each natural $k$ the smallest such $s=g(k)$ that all natural numbers $n$ can be written as sums of $s$ non-negative $k$-th powers, $n=x_1^k+\ldots+x_s^k$. In the XIX century the problem was modified as the quest of finding such minimal $s=G(k)$ that {\it almost all\/} $n$ can be expressed in this form. In the XX century this problem was further specified, as for finding such $G(k)$ {\it and\/} the precise list of exceptions. In the present talk I sketch the key steps in the solution of this problem, with a special emphasis on algebraic and computational aspects. I describe various connections of this problem, and its modifications, such as the rational Waring problem, the easier Waring problem, etc., with the current research in polynomial computer algebra,especially with identities, symbolic polynomials, etc. and promote several outstanding computational challenges.
KW - Waring problem
KW - easier Waring problem
KW - rational Waring problem
KW - polynomial identities
KW - sums of cubes
KW - sums of biquadrates
UR - https://pca-pdmi.ru/2020/files/53/Vavilov-PCA2020_new.pdf
M3 - Conference contribution
BT - International Conference Polynomial Computer Algebra 2020
A2 - Васильев, Николай
PB - Международный математический институт им. Эйлера
T2 - Polynomial Computer Algebra '2020
Y2 - 12 October 2020 through 17 October 2020
ER -
ID: 62862477