Duke Mathematical Journal

Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

Julius Borcea and Petter Brändén

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For (n×n)-matrices A and B, define η(A,B)=Sdet(A[S])det(B[S']), where the summation is over all subsets of {1,,n}, S' is the complement of S, and A[S] is the principal submatrix of A with rows and columns indexed by S. We prove that if A0 and B is Hermitian, then

(1) the polynomial η(zA,-B) has all real roots;

(2) the latter polynomial has as many positive, negative, and zero roots (counting multiplicities) as suggested by the inertia of B if A>0; and

(3) for 1in, the roots of η(zA[{i}'],-B[{i}']) interlace those of η(zA,-B).

Assertions (1)–(3) solve three important conjectures proposed by C. R. Johnson in the mid-1980s in [20, pp. 169, 170], [21]. Moreover, we substantially extend these results to tuples of matrix pencils and real stable polynomials. In the process, we establish unimodality properties in the sense of majorization for the coefficients of homogeneous real stable polynomials, and as an application, we derive similar properties for symmetrized Fischer products of positive-definite matrices. We also obtain Laguerre-type inequalities for characteristic polynomials of principal submatrices of arbitrary Hermitian matrices which considerably generalize a certain subset of the Hadamard, Fischer, and Koteljanskii inequalities for principal minors of positive-definite matrices. Finally, we propose Lax-type problems for real stable polynomials and mixed determinants

Article information

Duke Math. J., Volume 143, Number 2 (2008), 205-223.

First available in Project Euclid: 26 May 2008

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Zentralblatt MATH identifier

Primary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]
Secondary: 15A22: Matrix pencils [See also 47A56] 15A48 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10} 32A60: Zero sets of holomorphic functions 47B38: Operators on function spaces (general)


Borcea, Julius; Brändén, Petter. Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products. Duke Math. J. 143 (2008), no. 2, 205--223. doi:10.1215/00127094-2008-018. https://projecteuclid.org/euclid.dmj/1211819162

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