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1.
In this paper, we establish bounds on the degree of a symmetric polynomial p = p(x) = p(x
1,..., x
g
) (with real coefficients) in g noncommuting (nc) variables x
1,..., x
g
in terms of the “signature” of its Hessian
which is a polynomial in x and h = (h
1,..., h
g
) homogeneous of degree 2 in h. The bounds are obtained by exploiting the interplay between assorted representations for p(x) and p″(x)[h] that are developed in the paper. In particular, p″(x)[h] admits a representation of the form
where f
j
+
, f
j
−
are nc polynomials. Such representations are highly non-unique. However, there is a unique smallest number of positive (resp.,
negative) squares σ
±
min
required in an SDS decomposition of p″(x)[h]. Our main results yield the following corollary and a number of refinements.
Supported by a Jay and Renee Weiss Chair.
Partly supported by the NSF and the Ford Motor Co.
Partly supported by the NSF grants DMS-0140112 and DMS-0457504. 相似文献
2.
Suppose that X is a complex Banach space with the norm ‖·‖ and n is a positive integer with dim X ⩾ n ⩾ 2. In this paper, we consider the generalized Roper-Suffridge extension operator $
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)
on the domain $
\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }
$
\Omega _{p_1 ,p_2 , \ldots ,p_{n + 1} }
defined by
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)(x) = {*{20}c}
{\sum\limits_{j = 1}^n {\left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)} ^{\beta _j } (f'(x_1^* (x)))^{\gamma _j } x_1^* (x)x_j } \\
{ + \left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)^{\beta _{n + 1} } (f'(x_1^* (x)))^{\gamma _{n + 1} } \left( {x - \sum\limits_{j = 1}^n {x_1^* (x)x_j } } \right)} \\
$
\Phi _{n,\beta _2 ,\gamma _2 , \ldots ,\beta _{n + 1} ,\gamma _{n + 1} } (f)(x) = \begin{array}{*{20}c}
{\sum\limits_{j = 1}^n {\left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)} ^{\beta _j } (f'(x_1^* (x)))^{\gamma _j } x_1^* (x)x_j } \\
{ + \left( {\frac{{f(x_1^* (x))}}
{{x_1^* (x)}}} \right)^{\beta _{n + 1} } (f'(x_1^* (x)))^{\gamma _{n + 1} } \left( {x - \sum\limits_{j = 1}^n {x_1^* (x)x_j } } \right)} \\
\end{array}
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3.
Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a
0 + a
1
x + ... + a
k
x
k
is a k-degree polynomial with integral coefficients such that (p, a
0, a
1, ..., a
k
) = 1, for any integer m, we study the asymptotic property of
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