Classification of all noncommutative polynomials whose Hessian has negative signature one and a noncommutative second fundamental form
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: where f j + , f ℓ − are noncommutative polynomials. Let σ − min ( p ), the negative signatu...
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Veröffentlicht in: | Journal d'analyse mathématique (Jerusalem) 2009-05, Vol.108 (1), p.19-59 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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:
where
f
j
+
,
f
ℓ
−
are noncommutative polynomials. Let σ
−
min
(
p
), the negative signature of
p
, denote the minimum number of negative squares used in this representation; and let the
Hessian
of
p
be defined by the formula
In this paper, we classify all symmetric noncommutative polynomials
p
(
x
) such that
We also introduce the
relaxed Hessian
of a symmetric polynomial
p
of degree
d
via the formula
for λ, δ, ∈, ℝ and show that if this relaxed Hessian is positive semidefinite in a suitable and relatively innocuous way, then p has degree at most 2. Here the sum is over monomials
m
(
x
) in
x
of degree at most
d
− 1 and 1 ≤
j
≤
g
. This analysis is motivated by an attempt to develop properties of noncommutative real algebraic varieties pertaining to curvature, since, as will be shown elsewhere,
(appropriately restricted) plays the role of a noncommutative second fundamental form. |
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ISSN: | 0021-7670 1565-8538 |
DOI: | 10.1007/s11854-009-0017-0 |