Representation of Boolean Functions by Polynomials

This work considers such type of calculation of Boolean functions by polynomials as sign representation. This means that the value of function agrees with the sign of the polynomial on all inputs. The aim of this research is studying methods related to such representations and applying them for obtaining new theorems. In the main, it is bounds on degree and sparsity of the polynomial. My strategy in this research is to study Parity function in detail and to start from the results that achieved for this function. Parity function is in a sense a special function for sign representations. It can be sign represented only with polynomials that have high degree and sparsity but its value does not depend on the order of the variables. This feature has resulted in development of interesting methods of obtaining bounds by the case of Parity function. In the end it did not work out with transferring of the most complex and interesting theorems to other functions. But it did well with generalizing of a theorem which allows to obtain upper bounds on degree and sparsity of polynomials for sign representing of Parity function over any set of non-negative integers. I transferred this result to other functions and built a construction that allows to do the same for any Boolean function.