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Sign conditions for the existence of at least one positive solution of a sparse polynomial system. (English) Zbl 1456.13019

Let \(\mathcal A=\{a_1,\ldots,a_n\}\subset \mathbb R^d\), \(C=(c_{i,j})\in \mathbb R^{d\times n}\) and \(x=(x_1,\ldots,x_d)\). \(\mathcal A\) and \(C\) define a sparse generalized multivariate polynomial system \[f_i(x)=\sum_jc_{i,j}x^{a_j} \text{ , } i=1,\ldots,d.\] Sign conditions on the support \(\mathcal A\) and the coefficients \(C\) of the system \(f_1,\ldots,f_d\) are given that guarantee the existence of at least one positive real root, based on degree theory and Gale duality. In case of integer exponents algebraic conditions are given.

MSC:

13P15 Solving polynomial systems; resultants
14P05 Real algebraic sets

Software:

SINGULAR
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References:

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