An estimate for the number of terms in the Hilbert–Kamke problem

D. A. Mit'kin

Sbornik. Mathematics, Tome 57 (1987) no. 2, pp. 561-590

Résumé

Let $r(n)$ denote the smallest $s$ for which the system of equations \begin{equation} x^j_1+\dots+x^j_s=N_j\qquad(j=1,\dots,n) \end{equation} is solvable in nonnegative integers $x_1,\dots,x_s$ for all sufficiently large natural numbers $N_1,\dots,N_n$ which satisfy the following conditions: 1) the singular integral $\gamma=\gamma(N_1,\dots,N_n)$ of the system (1) satisfies the inequality $\gamma\geqslant c(n,s)>0$ (the order conditions). 2) the system of equations $\sum^n_{k=1}k^jt_k=N_j$ $(j=1,\dots,n)$ is solvable in integers $t_1,\dots,t_n$ (the arithmetic conditions). In 1937, K. K. Mardzhanishvili proved that $n^2\ll r(n)\leqslant n^42^{2n^2-n-2}$. G. I. Arkhipov has recently obtained upper and lower estimates for $r(n)$ having the same order of magnitude: $2^n-1\leqslant r(n)\leqslant3n^32^n-n$ $(n\geqslant5)$. In this paper, the upper estimate for $r(n)$ is reduced to \begin{equation} r(n)\leqslant\sum_{0\leqslant k\leqslant[\ln n/\ln2]}2^k(2^{[n/2^k]}-1)\qquad(n\geqslant12); \end{equation} in particular, the asymptotic formula $r(n)=2^n+O(2^{n/2})$ is obtained. It is conjectured that the estimate (2) is best possible. Bibliography: 20 titles.

Bibliographie

[1] Arkhipov G. I., “O znachenii osobogo ryada v probleme Gilberta–Kamke”, DAN SSSR, 259 (1981), 265–267 | MR | Zbl

[2] Arkhipov G. I., “O probleme Gilberta–Kamke”, Izv. AN SSSR. Ser. matem., 48:1 (1984), 3–52 | MR | Zbl

[3] Kamke E., “Verallgemeinerungen des Waring–Hilbertschen Satzes”, Math. Ann., 83 (1921), 85–112 | DOI | MR | Zbl

[4] Hilbert D., “Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n$ – ter Potenzen (Waringsches Problem)”, Math. Ann., 67 (1909), 281–300 | DOI | MR | Zbl

[5] Vinogradov I. M., Novyi metod v analiticheskoi teorii chisel, Tr. MIAN, 10, 1937 | Zbl

[6] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Tr. MIAN, 23, 1947 | MR | Zbl

[7] Vinogradov I. M., Metod trigonometricheskikh summ v teorii chisel, Nauka, M., 1980 | MR

[8] Mardzhanishvili K. K., “Ob odnovremennom predstavlenii $n$ chisel summami polnykh pervykh, vtorykh, ..., $n$-kh stepenei”, Izv. AN SSSR. Ser. matem., 1:4 (1937), 609–631

[9] Mardzhanishvili K. K., “O nekotorykh nelineinykh sistemakh uravnenii v tselykh chislakh”, Matem. sb., 33(75) (1953), 639–675 | Zbl

[10] Emelyanov G. V., “Ob odnoi sisteme diofantovykh uravnenii”, Uchenye zap. LGU, 19 (1950), 3–39 | MR

[11] Khua Lo-Ken, Additivnaya teoriya prostykh chisel, Tr. MIAN, 22, 1947 | MR | Zbl

[12] Gelfond A. O., Ischislenie konechnykh raznostei, Nauka, M., 1967 | MR

[13] Mardzhanishvili K. K., “Ob odnom osobom ryade”, Tr. MIAN, 142 (1976), 174–181 | Zbl

[14] Mordell G. L., “On a sum analogous to a Gauss's sum”, Quart. J. Math., 3 (1932), 161–167 | DOI | Zbl

[15] Karatsuba A. A., “Ob odnoi sisteme sravnenii”, Matem. zametki, 19:3 (1976), 389–392 | MR | Zbl

[16] Linnik Yu. V., “Nekotorye zamechaniya ob otsenkakh trigonometricheskikh summ”, UMN, 14:3 (1959), 153–160 | MR | Zbl

[17] Weil A., “On some exponential sums”, Proc. Nat. Acad. Sci. USA, 34:5 (1948), 204–207 | DOI | MR | Zbl

[18] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1972 | MR

[19] Hua L.-K., “On a generalized Waring problem. II”, J. Chinese Math. Soc., 2:2 (1940), 175–191 | MR | Zbl

[20] Nechaev V. I., “Otsenka polnoi ratsionalnoi trigonometricheskoi summy”, Matem. zametki, 17 (1975), 839–849 | Zbl