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In the literature, finite mixture models were described as linear combinations of probability distribution functions having the form $ f(x) = \Lambda \sum\limits_{i = 1}^n w_i f_i(x) $, $ x \in \mathbb{R} $, where $ w_i $ were positive weights, $ \Lambda $ was a suitable normalising constant, and $ f_i(x) $ were given probability density functions. The fact that $ f(x) $ is a probability density function followed naturally in this setting. Our question was: if we removed the sign condition on the coefficients $ w_i $, how could we ensure that the resulting function was a probability density function?The solution that we proposed employed an algorithm which allowed us to determine all zero-crossings of the function $ f(x) $. Consequently, we determined, for any specified set of weights, whether the resulting function possesses no such zero-crossings, thus confirming its status as a probability density function.In this paper, we constructed such an algorithm which was based on the definition of a suitable sequence of functions and that we called a generalized Budan-Fourier sequence; furthermore, we offered theoretical insights into the functioning of the algorithm and illustrated its efficacy through various examples and applications. Special emphasis was placed on generalized Gaussian mixture densities.

The problem of minimizing a polynomial function in several variables over Rn is considered and an algorithm is given. When the polynomial has a minimum the algorithm returns the global minimal value and finds at least one point in every connected component of the set of minimizers. A characterization of such points is given. When the polynomial does not have a minimum the algorithm computes its infimum. No assumption is made on the polynomial. [ABSTRACT FROM AUTHOR]