Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form |
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| Authors: | Harry Dym Jeremy M. Greene J. William Helton Scott A. McCullough |
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| Affiliation: | (1) Department of Mathematics, William Paterson University, Wayne, NJ 07470, USA |
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| Abstract: | Every symmetric polynomial p = p(x) = p(x 1,..., x g ) (with real coefficients) in g noncommuting variables x 1,..., x g can be written as a sum and difference of squares of noncommutative polynomials: | $
(SDS) p(x) = sumlimits_{j = 1}^{sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - sumlimits_{ell = 1}^{sigma _ - } {f_ell ^ - (x)^T f_ell ^ - (x)} ,
$
(SDS) p(x) = sumlimits_{j = 1}^{sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - sumlimits_{ell = 1}^{sigma _ - } {f_ell ^ - (x)^T f_ell ^ - (x)} ,
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