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**1 - 1**of**1**### Implementing Real Root Count with Polynomial Constraints for the Multivariate Case with one Constraint Polynomial

, 1995

"... Following [P. Pederson, et al. 94] and based on [H. Hong et al. 95] we develop several algorithms to compute multiplication tables and associated quadratic forms, which we compare in terms of efficiency. The results can immediately be applied to the general case of arbitrary many polynomials. 1 Int ..."

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Following [P. Pederson, et al. 94] and based on [H. Hong et al. 95] we develop several algorithms to compute multiplication tables and associated quadratic forms, which we compare in terms of efficiency. The results can immediately be applied to the general case of arbitrary many polynomials. 1 Introduction Problem specification We will use ¯ x as an abbreviation for x 1 ; : : : x r and let K denote any subfield of a real closed field R. Then we consider the following problem: given: A set of polynomials ff 1 ; : : : fn g such that f i 2 K[¯x] and the system f 1 = : : : = f n = 0 has finitely many solutions, and a constraint polynomial h 2 K[¯x]. find: The number of common real roots of the polynomials f i on which the constraint h ? 0 holds. Since h 0 is equivalent to (h = 0 h ? 0), we can reduce any polynomial constraint to the specified problem. Theory Similar to [P. Pederson, et al. 94] we introduce the following notations. Consider the finite dimensional vector space A ...