Geodesic
Relative completeness for functional systems of polynomials
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 967-977.

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For functional systems of polynomials with natural, integer and rational coefficients we solve the problem of completeness of sets, containing all monomials, and sets, containing all polynomials of one variable. 
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V. Sh. Darsalia. Relative completeness for functional systems of polynomials. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 4, pp. 967-977. http://geodesic.mathdoc.fr/item/FPM_2002_8_4_a1/

[1] Darsaliya V. Sh., “Usloviya polnoty dlya polinomov s naturalnymi, tselymi i ratsionalnymi koeffitsientami”, Fundam. i prikl. mat., 2:2 (1996), 365–374 | MR | Zbl

[2] Matiyasevich Yu. V., Desyataya problema Gilberta, Fizmatlit, M., 1993 | MR