Stability and complexity of mixed discriminants

1. Introduction and main results

Mixed discriminants were introduced by A.D. Alexandrov [Al38] in his work on mixed volumes and what was later called “the Alexandrov-Fenchel inequality”. Mixed discriminants generalize permanents and also found independent applications in problems of combinatorial counting, see, for example, Chapter 5 of [BR97], as well as in determinantal point processes [C+17], [KT12]. Recently, they made a spectacular appearance in the “mixed characteristic polynomial” introduced by Marcus, Spielman and Srivastava in their solution of the Kadison-Singer problem [M+15]. Over the years, the problem of computing or approximating mixed discriminants efficiently attracted some attention [GS02], [Gu05], [CP16].

In this paper, we establish some stability properties of mixed discriminants (the absence of zeros in certain complex domains) and, as a corollary, construct efficient algorithms to approximate the mixed discriminant of some sets of matrices. For example, we show that the mixed discriminant of $n \times n$ positive semidefinite matrices can be approximated within a relative error $ϵ > 0$ in quasi-polynomial $n^{O (ln n - ln ϵ)}$ time, provided the distance of each matrix to the identity matrix in the operator norm does not exceed some absolute constant $γ_{0} > 0$ . We also consider the case when all matrices have rank 2, shown to be $# P$ -hard by Gurvits [Gu05], and provide a quasi-polynomial approximation algorithm in a particular situation.

(1.1) Definitions and properties

Let $A_{1}, \dots, A_{n}$ be an $n$ -tuple of $n \times n$ complex matrices. The mixed discriminant of $A_{1}, \dots, A_{n}$ is defined by

D (A_{1}, \dots, A_{n}) = \frac{\partial^{n}}{\partial t_{1} \dots \partial t_{n}} det (t_{1} A_{1} + \dots + t_{n} A_{n}) .

The determinant in the right hand side is a homogeneous polynomial of degree $n$ in complex variables $t_{1}, \dots, t_{n}$ and $D (A_{1}, \dots, A_{n})$ is the coefficient of the monomial $t_{1} \dots t_{n}$ . It is not hard to see that $D (A_{1}, \dots, A_{n})$ is a polynomial of degree $n$ in the entries of $A_{1}, \dots, A_{n}$ : assuming that $A_{k} = (a_{i j}^{k})$ for $k = 1, \dots, n$ , we have

D (A_{1}, \dots, A_{n}) = \sum σ, τ \in S_{n} sgn σ n \prod i = 1 a_{i σ (i)}^{τ (i)} = \sum σ, τ \in S_{n} sgn (σ τ) n \prod i = 1 a_{τ (i) σ (i)}^{i},

where $S_{n}$ is the symmetric group of permutations of ${1, \dots, n}$ . It follows from (1.1.1) that $D (A_{1}, \dots, A_{n})$ is linear in each argument $A_{i}$ , and symmetric under permutations of $A_{1}, \dots, A_{n}$ , see, for example, Section 4.5 of [Ba16].

Mixed discriminants appear to be the most useful when the matrices $A_{1}, \dots, A_{n}$ are positive semidefinite real symmetric (or complex Hermitian), in which case $D (A_{1}, \dots, A_{n}) \geq 0$ . For a real $n$

-vector

x = (ξ_{1}, \dots, ξ_{n})

, let

x \otimes x

denote the

n \times n

matrix with the

(i, j)

-th entry equal

ξ_{i} ξ_{j}

. It is then not hard to see that

D (x_{1} \otimes x_{1}, \dots, x_{n} \otimes x_{n}) = {(det [x_{1}, \dots, x_{n}])}^{2},

where $[x_{1}, \dots, x_{n}]$ is the $n \times n$ matrix with columns $x_{1}, \dots, x_{n}$ , see, for example, Section 4.5 of [Ba16]. Various applications of mixed discriminants are based on (1.1.2). Suppose that $S_{1}, \dots, S_{n} \subset R^{n}$ are finite sets of vectors. Let us define

A_{k} = \sum x \in S_{k} x \otimes x for k = 1, \dots, n .

From the linearity of $D (A_{1}, \dots, A_{n})$ in each argument, we obtain

D (A_{1}, \dots, A_{n}) = \sum x_{1} \in S_{1}, \dots, x_{n} \in S_{n} {(det [x_{1}, \dots, x_{n}])}^{2} .

One combinatorial application of (1.1.3) is as follows: given a connected graph $G$ with $n$ vertices, color the edges of $G$ in $n - 1$ colors. Then the number of spanning trees containing exactly one edge of each color is naturally expressed as a mixed discriminant. More generally, this extends to counting “rainbow bases” in regular matroids with colored elements, cf. Chapter 5 of [BR97]. Another application of (1.1.3) is in determinantal point processes [C+17].

The mixed discriminant of (positive semidefinite) matrices generalizes the permanent of a (non-negative) matrix. Namely, given $n \times n$ diagonal matrices $A_{1}, \dots, A_{n}$ , we consider an $n \times n$ matrix $B$ whose $k$ -th row is the diagonal of $A_{k}$ . It is easy to see that

D (A_{1}, \dots, A_{n}) = per B,

where the permanent of $B$ is defined by

per B = \sum σ \in S_{n} n \prod i = 1 b_{i σ (i)} .

We note that if $A_{1}, \dots, A_{n}$ are positive semidefinite then $B$ is a non-negative matrix and that the permanent of any non-negative square matrix can be interpreted as the mixed discriminant of positive semidefinite matrices.

In their solution of the Kadison-Singer problem, Marcus, Spielman and Srivastava defined the mixed characteristic polynomial of $n \times n$ matrices $A_{1}, \dots, A_{m}$ by

p (x) = p_{A_{1}, \dots, A_{m}} (x) = m \prod k = 1 (1 - \frac{\partial}{\partial z_{k}}) det (x I + m \sum k = 1 z_{k} A_{k}) {∣ ∣}_{z_{1} = \dots = z_{m} = 0}

[M+15], see also [MS17]. The coefficients of $p (x)$ can be easily expressed as mixed discriminants, see Section 5.1. Finding efficiently the partition in the Weaver’s reformulation of the Kadison-Singer conjecture (the existence of such a partition is proven in [M+15]) reduces to bounding the roots of the mixed characteristic polynomial, which in turn makes computing the coefficients of the polynomial (which are expressed as mixed discriminants) of interest. It follows from our results that for any non-negative integer $c$ and $k$ , fixed in advance, one can approximate the coefficient of $x^{k}$ in quasi-polynomial time, provided $0 \leq m - n \leq c$ and the distance from each matrix $A_{i}$ to $I$ in the operator norm does not exceed an absolute constant $γ_{0} > 0$ .

Let $B = (b_{i j})$ be an $n \times n$ complex matrix. For a set $S \subset {1, \dots, n}$ , let $B_{S}$ be the $| S | \times | S |$ submatrix of $B$ consisting of the entries $b_{i j}$ with $i, j \in S$ . Gurvits [Gu05] noticed that computing the mixed discriminant of positive semidefinite matrices $A_{1}, \dots, A_{n}$ of rank 2 reduces to computing the sum

\sum S \subset {1, \dots, n} {(det B_{S})}_{S}^{2},

where for $S = \emptyset$

the corresponding term is equal to 1. Indeed, applying a linear transformation if necessary, we assume that for

k = 1, \dots, n

we have

A_{k} = e_{k} \otimes e_{k} + x_{k} \otimes x_{k}

, where

e_{1}, \dots, e_{n}

is the standard basis of

R^{n}

and

x_{1}, \dots, x_{n}

are some vectors in

R^{n}

. From the linearity of the mixed discriminant in each argument and (1.1.2), it follows that

D (e_{1} \otimes e_{1} + x_{1} \otimes x_{1}, \dots, e_{n} \otimes e_{n} + x_{n} \otimes x_{n}) = \sum S \subset {1, \dots, n} {(det B_{S})}_{S}^{2},

where $B = (b_{i j})$ is the Gram matrix of vectors $x_{1}, \dots, x_{n}$ , that is, $b_{i j} = ⟨ x_{i}, x_{j} ⟩$ and $⟨ \cdot, \cdot ⟩$ is the standard scalar product in $R^{n}$ , see [Gu05] for details. Computing a more general expression

\sum S \subset {1, \dots, n} {(det B_{S})}_{S}^{m}

for an integer $m \geq 2$ is of interest in discrete determinantal point processes [KT12].

As a ramification of our approach, we present a quasi-polynomial algorithm of $n^{O (ln n - ln ϵ)}$ complexity approximating (1.1.5) within relative error $0 < ϵ < 1$ provided $∥ B ∥ < ρ$ for any $ρ < 1$ , fixed in advance, where $∥ \cdot ∥$ is the operator norm.

(1.2) Computational complexity

Since mixed discriminants generalize permanents, they are at least as hard to compute exactly or to approximate as permanents. Moreover, it appears that mixed discriminants are substantially harder to deal with than permanents. It is shown in [Gu05] that it is a $# P$ -hard problem to compute $D (A_{1}, \dots, A_{n})$ even when $rank A_{k} = 2$ for $k = 1, \dots, n$ . In particular, computing (1.1.5) for a positive definite matrix $B$ and $m = 2$ is a $# P$

-hard problem. In contrast, the permanent of a matrix with at most 2 non-zero entries in each row is trivial to compute. A Monte Carlo Markov Chain algorithm of Jerrum, Sinclair and Vigoda

[J+04] approximates the permanent of a non-negative matrix in randomized polynomial time. Nothing similar is known or even conjectured to work for the mixed discriminant of positive semidefinite matrices. A randomized polynomial time algorithm from [Ba99] approximates the mixed discriminant of

n \times n

positive semidefinite matrices within a multiplicative factor of

c^{n}

for

c = 2 e^{γ - 1} \approx 1.31

, where

γ \approx 0.577

is the Euler constant. A deterministic polynomial time algorithm of [GS02] approximates the mixed discriminants of positive semidefinite matrices within a multiplicative factor of

e^{n} \approx (2.718)^{n}

. As is shown in Section 4.6 of [Ba16], for any

γ > 1

, fixed in advance, the scaling algorithm of [GS02] approximates the mixed discriminant

D (A_{1}, \dots, A_{n})

within a multiplicative factor of

n^{γ^{2}}

provided the largest eigenvalue of each matrix

A_{k}

is within a factor of

γ

of its smallest eigenvalue.

A combinatorial algorithm of [CP16] computes the mixed discriminant exactly in polynomial time for some class of matrices (of bounded tree width).

Our first result establishes the absence of complex zeros of $D (A_{1}, \dots, A_{n})$ if all $A_{i}$ lie sufficiently close to the identity matrix. In what follows, $∥ \cdot ∥$ denotes the operator norm a matrix, which in the case of a real symmetric matrix is the largest absolute value of an eigenvalue.

(1.3) Theorem

There is an absolute constant $γ_{0} > 0$ (one can choose $γ_{0} = 0.045$ ) such that if $Q_{1}, \dots, Q_{n}$ are $n \times n$ real symmetric matrices satisfying

∥ Q_{k} ∥ \leq γ_{0} for k = 1, \dots, n

then for $z_{1}, \dots, z_{n} \in C$ , we have

D (I + z_{1} Q_{1}, \dots, I + z_{n} Q_{n}) \neq 0 provided | z_{1} |, \dots, | z_{n} | \leq 1.

We note that under the conditions of the theorem, the mixed discriminant is not confined to any particular sector of the complex plane (in other words, the reasons for the mixed discriminant to be non-zero are not quite straightforward). For example, if $Q_{1} = \dots = Q_{n} = γ_{0} I$ , then $D (I + z Q_{1}, \dots, I + z Q_{n}) = n! (1 + γ_{0} z)^{n}$ rotates $Ω (n)$ times around the origin as $z$ ranges over the unit circle.

Applying the interpolation technique, see

[Ba16], [PR17], we deduce that the mixed discriminant

D (A_{1}, \dots, A_{n})

can be efficiently approximated if the matrices

A_{1}, \dots, A_{n}

are close to

I

in the operator norm. Let

Q_{1}, \dots, Q_{n}

be matrices satisfying the conditions of Theorem 1.3. Since

D (I + z_{1} Q_{1}, \dots, I + z_{n} Q_{n}) \neq 0

in the simply connected domain (polydisc)

| z_{1} |, \dots, | z_{n} | \leq 1

, we can choose a branch of

ln D (I + z_{1} Q_{1}, \dots, I + z_{n} Q_{n})

in that domain. It turns out that the logarithm of the mixed discriminant can be efficiently approximated by a low (logarithmic) degree polynomial.

(1.4) Theorem

For any $0 < ρ < 1$ there is a constant $c (ρ) > 0$ and for any $0 < ϵ < 1$ , for any positive integer $n$ , there is a polynomial

p = p_{ρ, n, ϵ} (Q_{1}, \dots, Q_{n}; z_{1}, \dots, z_{n})

in the entries of $n \times n$ real symmetric matrices $Q_{1}, \dots, Q_{n}$ and complex $z_{1}, \dots, z_{n}$ such that $deg p \leq c (ρ) (ln n - ln ϵ)$ and

| ln D (I + z_{1} Q_{1}, \dots, I + z_{n} Q_{n}) - p (Q_{1}, \dots, Q_{n}; z_{1}, \dots, z_{n}) | \leq ϵ

provided

∥ Q_{k} ∥ \leq γ_{0} for k = 1, \dots, n

where $γ_{0}$ is the constant in Theorem 1.4 and

| z_{1} |, \dots, | z_{n} | \leq ρ .

We show that the polynomial $p$ can be computed in quasi-polynomial $n^{O_{ρ} (ln n - ln ϵ)}$ time, where the implicit constant in the “ $O$ ” notation depends on $ρ$ alone. In other words, Theorem 1.4 implies that the mixed discriminant of positive definite matrices $A_{1}, \dots, A_{n}$ can be approximated within a relative error $ϵ > 0$ in quasi-polynomial $n^{O (ln n - ln ϵ)}$ time provided for each matrix $A_{k}$ , the ratio of any two eigenvalues is bounded by a constant $1 < γ < (1 + γ_{0}) / (1 - γ_{0})$ , fixed in advance. We note that the mixed discriminant of such $n$ -tuples can vary within an exponentially large multiplicative factor $γ^{n}$ .

Theorem 1.4 shows that the mixed discriminant can be efficiently approximated in some open domain in the space of $n$ -tuples of $n \times n$ symmetric matrices. A standard argument shows that unless $# P$ -hard problems can be solved in quasi-polynomial time, the mixed discriminant cannot be computed exactly in any open domain in quasi-polynomial time: if such a domain existed, we could compute the mixed discriminant exactly at any $n$ -tuple as follows: we choose a line through the desired $n$ -tuple and an $n$ -tuple in the domain; since the restriction of the mixed discriminant onto a line is a polynomial of degree $n$ , we could compute it by interpolation from the values at points in the domain.

We deduce from the Marcus - Spielman - Srivastava bound on the roots of the mixed characteristic polynomial [MS17] the following stability result for mixed discriminants.

(1.5) Theorem

Let $α_{0} \approx 0.278$ be the positive real solution of the equation $α e^{1 + α} = 1$ . Suppose that $Q_{1}, \dots, Q_{n}$ are $n \times n$ positive semidefinite matrices such that $Q_{1} + \dots + Q_{n} = I$ and $tr Q_{k} = 1$ for $k = 1, \dots, n$ . Then

D (I + z Q_{1}, \dots, I + z Q_{n}) \neq 0 for all z \in C such that | z | < \frac{α_{0} n}{4} .

As before, the interpolation argument produces the following algorithmic corollary.

(1.6) Theorem

For any $0 < ρ < α_{0} / 4 \approx 0.072$ , where $α_{0}$ is the constant of Theorem 1.5, there is a constant $c (ρ) > 0$ , and for any $0 < ϵ < 1$ , and any positive integer $n$ there is a polynomial

p = p_{ρ, n, ϵ} (Q_{1}, \dots, Q_{n}; z)

in the entries of $n \times n$ real symmetric matrices $Q_{1}, \dots, Q_{n}$ and complex $z$ such that $deg p \leq c (ρ) (ln n - ln ϵ)$ and

| ln D (I + z Q_{1}, \dots, I + z Q_{n}) - p (Q_{1}, \dots, Q_{n}; z) | \leq ϵ

provided $Q_{1}, \dots, Q_{n}$ are $n \times n$ positive semidefinite matrices such that

Q_{1} + \dots + Q_{n} = I, tr Q_{k} = 1 for k = 1, \dots, n

and $| z | \leq ρ n$ .

Again, the polynomial $p$ is constructed in quasi-polynomial $n^{O (ln n - ln ϵ)}$ time.

Some remarks are in order. An $n$ -tuple of $n \times n$ positive semidefinite matrices $Q_{1}, \dots, Q_{n}$ satisfying (1.6.1) is called doubly stochastic. Gurvits and Samorodnitsky [GS02] proved that an $n$ -tuple $A_{1}, \dots, A_{n}$ of $n \times n$ positive definite matrices can be scaled (efficiently, in polynomial times) to a doubly stochastic $n$ -tuple, that is, one can find an $n \times n$ matrix $T$ , a doubly stochastic $n$ -tuple $Q_{1}, \dots, Q_{n}$ , and positive real numbers $ξ_{1}, \dots, ξ_{n}$ such that $A_{k} = ξ_{k} T^{*} Q_{k} T$ for $k = 1, \dots, n$ , see also Section 4.5 of [Ba16] for an exposition. Then we have

D (A_{1}, \dots, A_{n}) = ξ_{1} \dots ξ_{n} (det T)^{2} D (Q_{1}, \dots, Q_{n})

and hence computing the mixed discriminant for any $n$ -tuple of positive semidefinite matrices reduces to that for a doubly stochastic $n$ -tuple. The $n$ -tuple $C = (n^{- 1} I, \dots, n^{- 1} I)$ naturally plays the role of the “center” of the set of all doubly stochastic $n$ -tuples. Let us contract the convex body of all doubly stochastic $n$ -tuples $X$ towards its center $C$ with a constant coefficient $γ < α_{0} / 4 \approx 0.07$ , $X ⟼ (1 - γ) C + γ X$ . Theorem 1.5 implies that the mixed discriminants of all contracted $n$ -tuples are efficiently (in quasi-polynomial time) approximable. In other words, there is “core” of the convex body of doubly stochastic $n$ -tuples, where the mixed discriminant is efficiently approximable, and that core is just a scaled copy (with a constant, small but positive, scaling coefficient) of the whole body.

Finally, we address the problem of computing (1.1.5). First, we prove the following stability result.

(1.7) Theorem

For an $n \times n$ complex matrix $B = (b_{i j})$ and a set $S \subset {1, \dots, n}$ , let $B_{S}$ be the submatrix of $B$ consisting of the entries $b_{i j}$ with $i, j \in S$ . For an integer $m \geq 1$ , we define a polynomial

ϕ_{B, m} (z) = \sum S \subset {1, \dots, n} {(det B_{S})}_{S}^{m} z^{| S |},

with constant term $1$ corresponding to $S = \emptyset$ . If $∥ B ∥ < 1$ then

ϕ_{B, m} (z) \neq 0 for any z \in C such % that | z | \leq 1.

Consequently, by interpolation we obtain the following result.

(1.8) Theorem

For any $0 < ρ < 1$ and any integer $m \geq 1$ there is a constant $c (ρ, m) > 0$ and for any $0 < ϵ < 1$ and integer $n$ there is a polynomial

p = p_{ρ, m, ϵ, n} (B)

in the entries of an $n \times n$ complex matrix $B$ such that $deg p \leq c (ρ, m) (ln n - ln ϵ)$ and

∣ ∣ ∣ ∣ ln ⎛ ⎝ \sum S \subset {1, \dots, n} {(det B_{S})}_{S}^{m} ⎞ ⎠ - p (B) ∣ ∣ ∣ ∣ \leq ϵ,

provided $B$ is an $n \times n$ matrix such that $∥ B ∥ < ρ$ .

The polynomial $p$ is constructed in quasi-polynomial $n^{O_{ρ, m} (ln n - ln ϵ)}$ time.

We prove Theorem 1.3 in Sections 2 and 3. We prove Theorem 1.4 in Section 4. In Section 5, we prove Theorems 1.5 and 1.6 and in Section 6, we prove Theorems 1.7 and 1.8.

2. Preliminaries

(2.1) From matrices to quadratic forms

Let $⟨ \cdot, \cdot ⟩$ be the standard scalar product in $R^{n}$ . With an $n \times n$ real symmetric matrix $Q$ we associate a quadratic form $q : R^{n} ⟶ R$ ,

q (x) = ⟨ x, Q x ⟩ for x \in R^{n} .

Given quadratic forms $q_{1}, \dots, q_{n} : R^{n} ⟶ R$ , we define their mixed discriminant by

D (q_{1}, \dots, q_{n}) = D (Q_{1}, \dots, Q_{n}),

where $Q_{k}$ is the matrix of $q_{k}$ . This definition does not depend on the choice of an orthonormal basis in $R^{n}$ (as long as the scalar product remains fixed): if we change the basis, the matrices change as $Q_{k} := U^{*} Q_{k} U$

for some orthogonal matrix

U

and all

k = 1, \dots, n

, and hence the mixed discriminant does not change.

The advantage of working with quadratic forms is that it allows us to define the mixed discriminant of the restriction of the forms onto a subspace. Namely, if $q_{1}, \dots, q_{m} : R^{n} ⟶ R$ are quadratic forms and $L \subset R^{n}$ is a subspace with $dim L = m$ , we make $L$ into a Euclidean space with the scalar product inherited from $R^{n}$ and define the mixed discriminant $D (q_{1} | L, \dots, q_{m} | L)$ for the restrictions $q_{k} : L ⟶ R$ .

We will use the following simple lemma.

(2.2) Lemma

Let $q_{1}, \dots, q_{n} : R^{n} ⟶ R$ be quadratic forms and suppose that

q_{n} (x) = n \sum k = 1 λ_{k} ⟨ u_{k}, x ⟩^{2},

where $λ_{1}, \dots, λ_{n}$ are real numbers and $u_{1}, \dots, u_{n} \in R^{n}$ are unit vectors. Then

D (q_{1}, \dots, q_{n}) = n \sum k = 1 λ_{k} D (q_{1} | u_{k}^{⊥}, \dots, q_{n - 1} | u_{k}^{⊥}),

where $u_{k}^{⊥}$ is the orthogonal complement to $u_{k}$ .

Demonstration Proof

This is Lemma 4.6.3 from [Ba16]. We give its proof here for completeness. By the linearity of the mixed discriminant in each argument, it suffices to check the formula when $q_{n} (x) = ⟨ u, x ⟩^{2}$ , where $u$ is a unit vector. Let $Q_{1}, \dots, Q_{n}$ be the matrices of $q_{1}, \dots, q_{n}$ in an orthonormal basis, where $u$ is the $n$ -th basis vector and hence $Q_{n}$ is the matrix where the $(n, n)$ -th entry is $1$ and all other entries are $0$ .

It follows from (1.1.1) that

D (Q_{1}, \dots, Q_{n}) = D (Q_{1}^{'}, \dots, Q_{n - 1}^{'}),

where $Q_{k}^{'}$ is the upper left $(n - 1) \times (n - 1)$ submatrix of $Q_{k}$ . We observe that $Q_{k}^{'}$ is the matrix of the restriction $q_{k} | u^{⊥}$ . ∎

(2.3) Comparing two restrictions

Let $q_{1}, \dots, q_{n - 1} : R^{n} ⟶ R$ be quadratic forms and let $u, v \in R^{n}$ be unit vectors (we assume that $u \neq \pm v)$ . We would like to compare $D (q_{1} | u^{⊥}, \dots, q_{n - 1} | u^{⊥})$ and $D (q_{1} | v^{⊥}, \dots, q_{n - 1} | v^{⊥})$ . Let $L = u^{⊥} \cap v^{⊥}$ , so $L \subset R^{n}$ is a subspace of codimension 2. Let us identify $u^{⊥}$ and $v^{⊥}$ with $R^{n - 1}$ as Euclidean spaces (we want to preserve the scalar product but do not worry about bases) in such a way that $L$ gets identified with $R^{n - 2} \subset R^{n - 1}$ . Hence the quadratic forms $q_{k} | u^{⊥}$ get identified with some quadratic forms $q_{k}^{u} : R^{n - 1} ⟶ R$ and the quadratic forms $q_{k} | v^{⊥}$ get identified with some quadratic forms $q_{k}^{v} : R^{n - 1} ⟶ R$ for $k = 1, \dots, n - 1$ .

We have

\begin{matrix} D (q_{1} | u^{⊥}, \dots, q_{n - 1} | u^{⊥}) = D (q_{1}^{u}, \dots, q_{n - 1}^{u}) and D (q_{1} | v^{⊥}, \dots, q_{n - 1} | v^{⊥}) = D (q_{1}^{v}, \dots, q_{n - 1}^{v}) . \end{matrix}

Besides

q_{k}^{u} | R^{n - 2} = q_{k}^{v} | R^{n - 2} for k = 1, \dots, n - 1.

Let us denote

r_{k} (x) = q_{k}^{u} (x) - q_{k}^{v} (x) for k = 1, \dots, n - 1.

Hence $r_{k} : R^{n - 1} ⟶ R$ are quadratic forms and by (2.3.1) we have $r_{k} (x) = 0$ for all $x \in R^{n - 2}$ . It follows then that

		$r_{k} (x) = ξ_{n - 1} ℓ_{k} (x) where x = (ξ_{1}, \dots, ξ_{n - 1})$

(2.4) Lemma

Suppose that $n \geq 3$ . Let $w, ℓ_{1}, ℓ_{2}, ℓ_{3} : R^{n} ⟶ R$ be linear forms. For $k = 1, 2, 3$ , let $r_{k} (x) = w (x) ℓ_{k} (x)$ be quadratic forms and let $q_{4}, \dots, q_{n} : R^{n} ⟶ R$ be some other quadratic forms. Then

D (r_{1}, r_{2}, r_{3}, q_{4}, \dots, q_{n}) = 0.

Demonstration Proof

Since the restriction of a linear form onto a subspace is a linear form on the subspace, repeatedly applying Lemma 2.2, we reduce the general case to the case of $n = 3$ , in which case the mixed discriminant in question is just $D (r_{1}, r_{2}, r_{3})$ . On the other hand, for all real $t_{1}, t_{2}, t_{3}$ we have

and hence

det (t_{1} r_{1} + t_{2} r_{2} + t_{3} r_{3}) = 0.

It follows by Definition 1.1 that $D (r_{1}, r_{2}, r_{3}) = 0$ . ∎

(2.5) Corollary

Suppose that $n \geq 2$ and let $q_{1}, \dots, q_{n - 1} : R^{n} ⟶ R$ be quadratic forms. Let $u, v \in R^{d}$ be unit vectors such that $u \neq - v$ and for $k = 1, \dots, n - 1$ , let us define quadratic forms $q_{k}^{u} : R^{n - 1} ⟶ R$ and $q_{k}^{v} : R^{n - 1} ⟶ R$ as in Section 2.3. Let $r_{k} = q_{k}^{u} - q_{k}^{v}$ for $k = 1, \dots, n - 1$ . Then

\begin{matrix} D (q_{1}^{u}, \dots, q_{n - 1}^{u}) = & D (q_{1}^{v}, \dots, q_{n - 1}^{v}) + n - 1 \sum i = 1 D (r_{i}, q_{k}^{v} : k \neq i) + \sum 1 \leq i < j \leq n - 1 D (r_{i}, r_{j}, q_{k}^{v} : k \neq i, j) . \end{matrix}

Moreover,

rank r_{k} \leq 2 for k = 1, \dots, n - 1.

If $n = 2$ , the second sum in the right hand side is empty.

Demonstration Proof

Since $q_{k}^{u} = q_{k}^{v} + r_{k}$ for $k = 1, \dots, n - 1$ , the proof follows by the linearity of the mixed discriminant in each argument and by Lemma 2.4. ∎

Finally, we will need a simple estimate.

(2.6) Lemma

Let $z$ and $w$ be complex numbers such that $| 1 - z | \leq δ$ and $| 1 - w | \leq τ$ for some $0 < δ, τ < 1$ . Then

R z w \geq 1 - δ - τ - δ τ .

Demonstration Proof

We write $z = 1 + a e^{i ϕ}$ and $w = 1 + b e^{i ψ}$ for some real $a, b, ϕ$ and $ψ$ such that $0 \leq a \leq δ$ and $0 \leq b \leq τ$ . Then

R z w = 1 + a cos ϕ + b cos ψ + a b cos (ϕ + ψ) \geq 1 - a - b - a b \geq 1 - δ - τ - δ τ .

∎

3. Proof of Theorem 1.3

We prove Theorem 1.3 by induction on $n$ . Following Section 2, we associate with (now complex) matrices $I + z_{k} Q_{k}$ (now complex-valued) quadratic forms $p_{k} (x) = ∥ x ∥^{2} + z_{k} q_{k} (x)$ where $q_{k} : R^{n} ⟶ R$ is the quadratic form with matrix $Q_{k}$ . If $L \subset R^{n}$ is a subspace then the restriction of $p_{k}$ onto $L$ is just $∥ x ∥^{2} + z_{k} (q_{k} | L)$ , where $q_{k} | L$ is the restriction of $q_{k}$ onto $L$ . The induction is based on the following two lemmas.

(3.1) Lemma

Let us fix $0 < δ, τ < 1$ such that $δ + τ + δ τ < 1$ . Let $q_{1}, \dots, q_{n} : R^{n} ⟶ R$ , $n \geq 2$ , be quadratic forms and let $z_{1}, \dots, z_{n}$ be complex numbers such that $| z_{1} |, \dots, | z_{n} | \leq 1$ . Let us define

p_{k} (x) = ∥ x ∥^{2} + z_{k} q_{k} (x), p_{k} : R^{n} ⟶ C for k = 1, \dots, n

and suppose that the following conditions hold:

Then for any unit vector $v \in R^{n}$ , we have

| D (p_{1}, \dots, p_{n}) | \geq n (1 - δ - τ - τ δ) ∣ ∣ D (p_{1} | v^{⊥}, \dots, p_{n - 1} | v^{⊥}) ∣ ∣ .

Demonstration Proof

We have

q_{n} (x) = n \sum k = 1 λ_{k} ⟨ u_{k}, x ⟩^{2},

where $u_{1}, \dots, u_{n}$

are the orthonormal eigenvectors of

q_{n}

and

λ_{1}, \dots, λ_{n}

are the corresponding eigenvalues. In particular, from condition (2) of the lemma, we have

| λ_{k} | \leq δ for k = 1, \dots, n .

Since

n \sum k = 1 ⟨ u_{k}, x ⟩^{2} = ∥ x ∥^{2},

from Lemma 2.2, we obtain by the linearity of the mixed discriminant

D (p_{1}, \dots, p_{n}) = n \sum k = 1 (1 + z_{n} λ_{k}) D (p_{1} | u_{k}^{⊥}, \dots, p_{n - 1} | u_{k}^{⊥}) .

Let us choose a unit vector $v \in R^{n}$ . Then

D (p_{1} | u_{k}^{⊥}, \dots, p_{n - 1} | u_{k}^{⊥}) = (1 + α (u_{k}, v)) D (p_{1} | v^{⊥}, \dots, p_{n - 1} | v^{⊥})

for some $α (u_{k}, v) \in C$ such that $| α (u_{k}, v) | \leq τ$ .

Combining (3.1.1) and (3.1.2), we get

D (p_{1}, \dots, p_{n}) = D (p_{1} | v^{⊥}, \dots, p_{n - 1} | v^{⊥}) n \sum k = 1 (1 + z_{n} λ_{k}) (1 + α (u_{k}, v)) .

From Lemma 2.6,

R (n \sum k = 1 (1 + z_{n} λ_{k}) (1 + α (u_{k}, v))) \geq n (1 - δ - τ - δ τ) > 0

and hence

∣ ∣ ∣ ∣ n \sum k = 1 (1 + z_{n} λ_{k}) (1 + α (u_{k}, v)) ∣ ∣ ∣ ∣ \geq n (1 - δ - τ - δ τ) .

The proof then follows from (3.1.3). ∎

(3.2) Lemma

Let us fix $0 < δ, μ < 1$ such that

4 δ μ^{- 1} + 8 δ^{2} μ^{- 2} < 1.

Let $q_{1}, \dots, q_{n - 1} : R^{n} ⟶ R$ , $n \geq 2$ , be quadratic forms and let $z_{1}, \dots, z_{n - 1}$ be complex numbers such that $| z_{1} |, \dots, | z_{n - 1} | \leq 1$ . Let us define

p_{k} (x) = ∥ x ∥^{2} + z_{k} q_{k} (x), p_{k} : R^{n} ⟶ C for k = 1, \dots, n - 1

and suppose that the following conditions hold:

Then for any two unit vectors $u, v \in R^{n}$ , we have

D (p_{1} | u^{⊥}, \dots, p_{n - 1} | u^{⊥}) = (1 + α (u, v)) D (p_{1} | v^{⊥}, \dots, p_{n - 1} | v^{⊥})

for some $α (u, v) \in C$ such that

| α (u, v) | \leq 4 δ μ^{- 1} + 8 δ^{2} μ^{- 2} .

DemonstrationProof

As in Section 2.3, let us construct the quadratic forms $q_{k}^{u}, q_{k}^{v} : R^{n - 1} ⟶ R$ for $k = 1, \dots, n - 1$ and the corresponding forms $p_{k}^{u}, p_{k}^{v} : R^{n - 1} ⟶ C$ . Clearly,

p_{k}^{u} (x) = ∥ x ∥^{2} + z_{k} q_{k}^{u} (x) and p_{k}^{v} (x) = ∥ x ∥^{2} + z_{k} q_{k}^{v} (x) for k = 1, \dots, n - 1

and

\begin{matrix} D (p_{1} | u^{⊥}, \dots, p_{n - 1} | u^{⊥}) = D (p_{1}^{u}, \dots, p_{n - 1}^{u}) and D (p_{1} | v^{⊥}, \dots, p_{n - 1} | v^{⊥}) = D (p_{1}^{v}, \dots, p_{n - 1}^{v}) . \end{matrix}

Let

r_{k} (x) = q_{k}^{u} (x) - q_{k}^{v} (x) and hence p_{k}^{u} (x) - p_{k}^{v} (x) = z_{k} r_{k} (x) for k = 1, \dots, n - 1.

From condition (2) of the lemma, we have

| r_{k} (x) | \leq 2 δ ∥ x ∥^{2} for k = 1, \dots, n - 1.

From Corollary 2.5,

	$D (p_{1}^{u}, \dots, p_{n - 1}^{u}) =$	$D (p_{1}^{v}, \dots, p_{n - 1}^{v}) + n - 1 \sum i = 1 z_{i} D (r_{i}, p_{k}^{v} : k \neq i)$
		$+ \sum 1 \leq i < j \leq n - 1 z_{i} z_{j} D (r_{i}, r_{j}, p_{k}^{v} : k \neq i, j)$

and

rank r_{k} \leq 2 for k = 1, \dots, n - 1.

If $n = 2$ then the second sum is absent in the right hand side of (3.2.2).

We can write

r_{k} (x) = λ_{k 1} ⟨ w_{k 1}, x ⟩^{2} + λ_{k 2} ⟨ w_{k 2}, x ⟩^{2}

where $λ_{k 1}$ and $λ_{k 2}$ are eigenvalues of $r_{k}$ with the corresponding unit eigenvectors $w_{k 1}$ and $w_{k 2}$ . By (3.2.1) we have

| λ_{k 1} |, | λ_{k 2} | \leq 2 δ for k = 1, \dots, n - 1.

Applying Lemma 2.2, we obtain

where the final inequality follows by condition (1) of the lemma. For $n = 2$ we just have

| D (r_{1}) | \leq 2 δ < \frac{4 δ}{μ} .

Similarly, if $n \geq 3$ , for every $1 \leq i < j \leq n - 1$ , we obtain

where each of the four terms in the right hand side is $0$ if the corresponding intersection of subspaces $w_{i 1}^{⊥}, w_{i 2}^{⊥}, w_{j 1}^{⊥}$ or $w_{j 2}^{⊥}$ fails to be $(n - 3)$ -dimensional. Hence we get

Stability and complexity of mixed discriminants

Some facts on Permanents in Finite Characteristics

Computing permanents of complex diagonally dominant matrices and tensors

Deterministic computation of the characteristic polynomial in the time of matrix multiplication

The infinity norm bounds and characteristic polynomial for high order RK matrices

On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices

Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices

Approximating the Permanent of a Random Matrix with Vanishing Mean

1. Introduction and main results

(1.1) Definitions and properties

(1.2) Computational complexity

(1.3) Theorem

(1.4) Theorem

(1.5) Theorem

(1.6) Theorem

(1.7) Theorem

(1.8) Theorem

2. Preliminaries

(2.1) From matrices to quadratic forms

(2.2) Lemma

Demonstration Proof

(2.3) Comparing two restrictions

(2.4) Lemma

Demonstration Proof

(2.5) Corollary

Demonstration Proof

(2.6) Lemma

Demonstration Proof

3. Proof of Theorem 1.3

(3.1) Lemma

Demonstration Proof

(3.2) Lemma

DemonstrationProof

Stability and complexity of mixed discriminants

Related Research

Some facts on Permanents in Finite Characteristics

Computing permanents of complex diagonally dominant matrices and tensors

Deterministic computation of the characteristic polynomial in the time of matrix multiplication

The infinity norm bounds and characteristic polynomial for high order RK matrices

On maximum volume submatrices and cross approximation for symmetric semidefinite and diagonally dominant matrices

Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices

Approximating the Permanent of a Random Matrix with Vanishing Mean

1. Introduction and main results

(1.1) Definitions and properties

(1.2) Computational complexity

(1.3) Theorem

(1.4) Theorem

(1.5) Theorem

(1.6) Theorem

(1.7) Theorem

(1.8) Theorem

2. Preliminaries

(2.1) From matrices to quadratic forms

(2.2) Lemma

Demonstration Proof

(2.3) Comparing two restrictions

(2.4) Lemma

Demonstration Proof

(2.5) Corollary

Demonstration Proof

(2.6) Lemma

Demonstration Proof

3. Proof of Theorem 1.3

(3.1) Lemma

Demonstration Proof

(3.2) Lemma

DemonstrationProof