Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

Abstract This article addresses the problem of computing an upper bound degree d a polynomial solution P ( x ) algebraic difference equation form G − τ 1 , … s + 0 = when such with coefficients in field K characteristic zero exists and where is non-linear s-variable [ ] . It will be shown that if quadratic constant then one can construct countable family polynomials f l u there (minimal) index being non-zero polynomial, its roots or ≤ deg ⁡ Moreover, existence proven for real numbers. These results are based on properties modules generated by special families homogeneous symmetric polynomials. A sufficient condition similar arbitrary total variable as well.