Abstract
We show that positivity (≥0) on mathbb{R}_{+}^{n} and on mathbb{R}^{n} of real symmetric polynomials of degree at most p in nge 2 variables is solvable by algorithms running in polynomial time in the number n of variables. For real symmetric quartics, we find discriminants which lead to the efficient algorithms QE4+ and QE4 running in O(n) time. We describe the Maple implementation of both algorithms, which are then used not only for testing concrete inequalities (with given numerical coefficients and number of variables), but also for proving symbolic inequalities.
Highlights
1 Introduction In this paper we deal with special cases of the quantifier elimination problems f (x) ≥ 0 ∀x ∈ Rn+, f (x) ≥ 0 ∀x ∈ Rn, for polynomials f ∈ R[X1, . . . , Xn], which will be referred to as QE+(f ) and QE(f ), respectively
Positive semidefinite symmetric fourth order tensors play an important role in continuum mechanics and in diffusion weighted magnetic resonance image processing (DW-MRI, for investigating the complex microstructure of the cerebral white matter in-vivo and non-invasively); positive semidefiniteness of such tensors is equivalent to the condition QE(f ) for a homogeneous symmetric quartic f
Since the resulting algorithms QE4+ and QE4 test the signs of finitely many discriminants (Theorems 4 and 8), the usual requirements are fulfilled
Summary
In this paper we deal with special cases of the quantifier elimination problems f (x) ≥ 0 ∀x ∈ Rn+, f (x) ≥ 0 ∀x ∈ Rn, for polynomials f ∈ R[X1, . . . , Xn], which will be referred to as QE+(f ) and QE(f ), respectively. Xn], which will be referred to as QE+(f ) and QE(f ), respectively. In this paper we deal with special cases of the quantifier elimination problems f (x) ≥ 0 ∀x ∈ Rn+, f (x) ≥ 0 ∀x ∈ Rn, for polynomials f ∈ R[X1, . The latter is related to Hilbert’s 17th problem on the possibility of writing positive semidefinite real polynomials as sums of squares, and to Artin’s solution to it, according to which the polynomials satisfying QE(f ) are precisely those which are sums of finitely many squares of rational functions. Following the notations from [7,8,9], let us consider the vector space [n] p of all real symmetric polynomials of degree at most p ∈ N in n ∈ N∗ indeterminates and the subspace
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