Algebraic polynomials and moments of stochastic integrals

We propose an algebraic method for proving estimates on moments of stochastic integrals. The method uses qualitative properties of roots of algebraic polynomials from certain general classes. As an application, we give a new proof of a variation of the Burkholder-Davis-Gundy inequality for the case of stochastic integrals with respect to real locally square integrable martingales. Further possible applications and extensions of the method are outlined.