Analytical aspects of Liouville-type equations with singular by Tarantello G.

By Tarantello G.

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Hilbert [Ill, who posed a further problem of simultaneous representation of several numbers by finite sums of powers of natural numbers with exponents 1 , 2 , .. , n , or in other words, the solvability of the system of diophantine equations in natural number unknowns x1, . . , xk. This problem subsequently became known as the "Hilbert-Kamke problem", with the corresponding function G l (n). If we replace the natural numbers by primes, then we have the Goldbach problem and the Hilbert-Kamke problem in primes, respectively (to the latter of which we refer later).

I. Arkhipov [2], who found the necessary and sufficient order conditions. The second type of necessary conditions is of arithmetical nature found by K. K. Mardzhanischvili [13], who, using Vinogradov's method [15][22], showed that for an appropriate k = k ( n ) , these conditions are also 29 On the Hilbert-Kamke and the Vinogradov problems sufficient. G. I. Arkhipov [2] showed that these arithmetical conditions can be expressed as the solvability of the system of linear equations in integers t l , .

N,), N1 + oo,belongs to the (7,E)cone if its entries satisfy the conditions where 7 = ( y l , . . , y,), 71,.. , are some positive constants and E is a small positive constant. An n-tuple belonging t o the (7,&)-coneis said 31 On the Hilbert-Kamke and the Vznogradov problems to satisfy the real solvability condition if there exists a number P = P(7) such that for each N1 P, the system of equations > < < < < < < < xi 1, 1 i 5 k, and the Jacobi is solvable in real numbers xi, 0 matrix (1 m n,1 s k) m of the solution X I , .

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