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In this paper, we first introduce the concept of Laurent differentially essential systems and give a criterion for a Laurent differential polynomial system to be Laurent differentially essential in terms of its support matrix. Then, the sparse differential resultant for a Laurent differentially essential system is defined, and its basic properties are proved. In particular, order and degree bounds for the sparse differential resultant are given. Based on these bounds, an algorithm to compute the sparse differential resultant is proposed, which is single exponential in terms of the Jacobi number and the size of the system. [ABSTRACT FROM AUTHOR]

Abstract: Efficient characteristic set methods for computing zeros of polynomial equation systems in a finite field are proposed. The concept of proper triangular sets is introduced and an explicit formula for the number of zeros of a proper and monic triangular set is given. An improved zero decomposition algorithm is proposed to reduce the zero set of an equation system to the union of zero sets of monic proper triangular sets. The bitsize complexity of this algorithm is shown to be for Boolean polynomials, where is the number of variables and is the number of equations. We also give a multiplication free characteristic set method for Boolean polynomials, where the sizes of the polynomials occurred during the computation do not exceed the sizes of the input polynomials and the bitsize complexity of algorithm is for input polynomials with variables and degree . The algorithms are implemented in the case of Boolean polynomials and extensive experiments show that they are quite efficient for solving certain classes of Boolean equations raising from stream ciphers. [Copyright &y& Elsevier]

In this paper, the concept of sparse difference resultant for a Laurent transformally essential system of difference polynomials is introduced and a simple criterion for the existence of sparse difference resultant is given. The concept of transformally homogenous polynomial is introduced and the sparse difference resultant is proved to be transformally homogenous. It is shown that the vanishing of the sparse difference resultant gives a necessary condition for the corresponding difference polynomial system to have non-zero solutions. Order and degree bounds for the sparse difference resultant are given. Based on these bounds, an algorithm to compute the sparse difference resultant is proposed, which is single exponential in terms of the number of variables, the Jacobi number, and the size of the Laurent transformally essential system. Furthermore, the precise order and degree, a determinant representation, and a Poisson-type product formula for the difference resultant are given. [ABSTRACT FROM AUTHOR]