The three-matrix coefficient-minor stability conjecture

Let $D_1,D_2,D_3$ be positive definite matrices, and let $M$ be the matrix whose entries are the coefficients of $y$ and $z$ in

\det(I+xD_1+yD_2+zD_3).

A minor means the determinant of any finite square submatrix of $M$ . The three-matrix coefficient-minor stability conjecture. Every minor of $M$ is a stable polynomial in $x$ . The source calls this a surprising conjecture and does not establish it.

Progress summary

Open

A posted calculation claims a concrete counterexample disproves the conjecture in three dimensions, but nobody has independently checked it.

The conjecture asserts that every coefficient minor arising from three positive definite matrices is stable as a polynomial in $x$ . The supplied source describes this as surprising and does not prove it.

Posted attempt

A reader-written calculation gives three strictly positive definite $3\times3$ integer matrices and an explicit coefficient minor $\Delta(x)=-4(3076x^3+3012x^2-99x-440)$ . Since $\Delta(0)>0$ and $\Delta(1)<0$ , it claims a zero $x_0\in(0,1)$ , hence a failure of stability and a complete counterexample. The calculation has not been independently verified.

Current status (as of August 2026): A reader-posted counterexample claims to disprove the conjecture, but no independent verification was found, so the conjecture remains mathematically unconfirmed.

Sources & referencesView supporting material

Primary source

Steve Fisk, “Polynomials, roots, and interlacing”, arXiv:math/0612833 (2008).

1 week ago

The conjecture fails already for three strictly positive definite $3\times3$ integer matrices. Take

D_1= \begin{pmatrix} 1&0&0\\ 0&2&0\\ 0&0&1 \end{pmatrix}, \qquad D_2= \begin{pmatrix} 13&9&7\\ 9&7&5\\ 7&5&4 \end{pmatrix}, \qquad D_3= \begin{pmatrix} 5&7&8\\ 7&11&12\\ 8&12&14 \end{pmatrix}.

Their respective lists of leading principal minors are

(1,2,2),\qquad(13,10,2),\qquad(5,6,4).

Hence all three matrices are positive definite by Sylvester's criterion.

Write

M_{ij}(x)=[y^iz^j]\det(I+xD_1+yD_2+zD_3).

The submatrix with row and column indices $0,1,2$ is

\begin{pmatrix} 2x^3+5x^2+4x+1&49x^2+79x+30&28x+22\\ 41x^2+65x+24&254x+164&64\\ 19x+16&62&0 \end{pmatrix}.

Its determinant equals

\Delta(x) =-4\left(3076x^3+3012x^2-99x-440\right).

But

\Delta(0)=1760>0, \qquad \Delta(1)=-22196<0.

The intermediate value theorem therefore gives a real root $x_0\in(0,1)$ . In particular, $\Delta$ has a zero in the open right half-plane and is not stable.

Thus a coefficient minor of $\det(I+xD_1+yD_2+zD_3)$ need not be stable, even when every $D_i$ is strictly positive definite.

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Shivam Patel · 2 days ago