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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| Institution: | (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:
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$
(SDS) p(x) = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)^T f_\ell ^ - (x)} ,
$
(SDS) p(x) = \sum\limits_{j = 1}^{\sigma _ + } {f_j^ + (x)^T f_j^ + (x)} - \sum\limits_{\ell = 1}^{\sigma _ - } {f_\ell ^ - (x)^T f_\ell ^ - (x)} ,
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| Keywords: | |
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