A nonspectrahedral hyperbolicity cone

1 Introduction

A homogeneous polynomial p\in\R[x_1,\ldots,x_m] of degree d\geq1 is hyperbolic with respect to e\in\R^m if p(e)\ne0 and every root of t\mapsto p(te-x) is real for every x\in\R^m. Write these roots, with multiplicity, as \lambda_1(x),\ldots,\lambda_d(x). The closed hyperbolicity cone is

We write \Sym^N for the real symmetric N\times N matrices and A\succeq0 (respectively, A\succ0) for positive semidefiniteness (respectively, positive definiteness). A cone is spectrahedral if it has the form

for some finite N\geq1. The geometric Generalized Lax conjecture asserts that every hyperbolicity cone is spectrahedral. We give a counterexample.

1.1 The explicit polynomial

All indices in \{1,2,3\} in the following construction are cyclic. For z,y\in\R^3, let

This is the coefficient-one Choi–Lam biquadratic form [4, Section 4]. Define Q(y)\in\Sym^3 by

so that b(z,y)=z^\top Q(y)z. For a\in\R, r\in\R^3 and T\in\Sym^3, define the linear map \Phi_y:\Sym^4\to\Sym^4 by

The dependence on y is homogeneous quadratic. If \adj X denotes the adjugate of X, set

This formula is polynomial even when X is singular. The ambient space has dimension 10+10+3=23.

The conclusion concerns the cone itself, so it allows every possible defining pencil and every finite matrix size. The compact formula for p is convenient for the proof, but one determinant factor can be removed.

We prove the degree reduction in Section 7, after the obstruction argument. We make no claim that the dimension or degree is minimal. The real symmetric convention also covers Hermitian pencils: for a Hermitian matrix H=A+iB, positivity is equivalent to positivity of the real symmetric matrix \left(\begin{smallmatrix}A&-B\\B&A\end{smallmatrix}\right).

1.2 Historical context

Gårding’s theory establishes the convexity of hyperbolicity cones [6]. The original Lax conjecture asks for a definite determinantal representation of homogeneous hyperbolic polynomials in three variables [12]. Helton and Vinnikov proved the corresponding definite determinantal representation theorem [9]; Lewis, Parrilo, and Ramana established its equivalence with Lax’s formulation [13]. In higher dimension, Brändén constructed a real-zero polynomial none of whose positive powers admits a definite determinantal representation [2, Theorem 3.3]. That obstruction does not exclude a different determinant, with additional factors, from defining the same cone. The geometric conjecture concerns this remaining possibility. Kummer exhibited the distinction concretely: the cone of the specialized Vámos polynomial has a 7\times7 representation, although no positive power of that polynomial has a definite determinantal representation [10, Section 3]. Raghavendra, Ryder, Srivastava, and Weitz proved exponential lower bounds on the matrix size of spectrahedral representations for suitable hyperbolicity cones [16]. Recent positive results include Netzer’s theorem for hyperbolic cubics in five variables [14, Theorem 3.9].

Kummer and Netzer prove spectrahedrality for cones of strictly hyperbolic polynomials [11, Theorem 1]. Here strictly hyperbolic means that p(te-x) has distinct roots whenever x\notin\R e. The polynomial (1.4) fails this hypothesis: at w=(I_4,0,0)\notin\R e it gives

For the reduced polynomial of Corollary 1.2, the same line gives q(te-w)=(t-1)^{12}t^4. Thus neither defining polynomial satisfies the strict-case hypothesis.

A spectrahedral shadow is a linear image of a set defined by an affine real symmetric linear matrix inequality. Netzer and Sanyal proved that a hyperbolicity cone is a spectrahedral shadow when every nonzero boundary point is a smooth point of the defining polynomial [15, Theorem 1.1]. Our result concerns a direct representation in the original variables; it makes no assertion about representations using auxiliary variables.

The scalar ingredient is the coefficient-one Choi–Lam biquadratic form [4, Section 4], which reduces the coefficient of the extra cyclic squared monomials from two in Choi’s earlier nonnegative form [3] to one. Its inability to dominate a nonzero bilinear square follows from the extremality proved in [4, Theorem 4.4]; we give a short proof of precisely this property. It is called weak extremality by Blekherman, Raiţă, Shankar, and Sinn [1, Definition 2.7], whose characterization uses Taylor expansions of dominated squares at zeros [1, Theorem 3.2]. The correspondence between nonnegative biquadratic forms and positive maps on real symmetric matrices is classical; see [8, Section 3]. Positive maps have also been used to obtain certificates of conic polynomial stability [5]. Here the construction turns the scalar obstruction into an obstruction to every finite pencil describing one hyperbolicity cone. A related general strategy appears in Scheiderer’s local sums-of-squares obstructions to spectrahedral shadows [17, Section 4]. Our proof extracts bilinear square minorants from a differentiated operator-norm identity; it does not invoke a criterion for shadows.

González Nevado states a positive solution of the geometric conjecture in [7, Theorem 53, pp. 16–17]. Appendix A records a counterexample to the path-containment assertion used in that argument. The construction and proof in the present paper are independent of that comparison.

1.3 Proof strategy

The scalar obstruction is particularly rigid: b is nonnegative and nonzero, but every bilinear form \ell satisfying \ell(z,y)^2\leq b(z,y) for all z,y is zero. Section 2 gives a short proof using zeros and curves along which b vanishes to fourth order.

The map \Phi_y translates this form into a matrix inequality. In Section 3 we prove hyperbolicity and identify the slice of K with X\succ0 as

The remaining argument assumes an arbitrary pencil defining K and extracts a bilinear square dominated by b. First, Section 4 rescales the pencil along (X,s^2Z,sy) and obtains an equivalent block matrix inequality. Section 5 then sends X^{-1} and Z^{-1} toward rank-one projections; the resulting comparison yields an exact operator-norm identity for kernel projections of the pencil blocks. These projections are parametrized by the unit sphere in \R^4. Finally, Section 6 differentiates one kernel projection on a neighborhood of constant rank in that sphere. At a suitable base point, the three-dimensional tangent space can be identified with all the z-variables, so this local identity controls b for every z and y. Each entry of the resulting matrix is a bilinear form whose square is bounded by b, giving a contradiction.

The mechanism connecting the biquadratic form to the geometry of the cone is the rescaling and subsequent first variation of these kernel projections. It applies to an arbitrary hypothetical pencil without comparing its determinant with p. All matrix norms below are Euclidean operator norms, denoted by \norm{\,\cdot\,}_{\op}; vector norms are Euclidean.

2 A biquadratic form with no dominated square

The geometric argument will produce bilinear forms whose squares are bounded by the Choi–Lam form b in (1.2). We prove directly the particular consequence of its classical extremality [4, Theorem 4.4] that the geometric argument needs: every such bilinear form must vanish. This property is also called weak extremality [1, Definition 2.7].

3 Hyperbolicity and two cone slices

We first prove that the polynomial in (1.4) is hyperbolic. We then identify the two slices of its closed cone that will constrain any representing pencil.

For u=(u_0,u') and v=(v_0,v') in \R^4, the block formula (1.3) gives

Indeed, the three terms on expanding the middle expression are v_0^2u'^\top Q(y)u', -2u_0v_0u'^\top Q(y)v', and u_0^2v'^\top Q(y)v', exactly the terms from the block formula. Lemma 2.1 therefore implies that \Phi_y(vv^\top)\succeq0. Every positive semidefinite real symmetric matrix is a nonnegative linear combination of such rank-one matrices. Thus \Phi_y is a positive linear map: it preserves positive semidefiniteness, and hence the order \preceq. The same formula also gives

so in particular \Phi_0=0 and \Phi_{-y}=\Phi_y. This rank-one verification is the real-symmetric positive-map correspondence for biquadratic forms [8, Section 3] applied to the explicit map (1.3).

4 What a semidefinite representation would imply

The two slices in Proposition 3.2 constrain every pencil representing K. We now turn those constraints into an exact block matrix inequality whose off-diagonal block is linear in y. A positive linear map preserves positive semidefiniteness; it is called unital if it sends the identity to the identity.

The maps and their finite dimensions are now fixed by the hypothetical pencil. We next recover the scalar form b from (4.1); this will force a nonzero bilinear square dominated by b and contradict its defining obstruction.

5 Rank-one limits and a norm identity

Let D,E,B be the maps supplied by Proposition 4.1. Thus D and E are positive maps with output sizes a,c\geq1 and are unital, meaning D(I_4)=I_a and E(I_4)=I_c, and B(y) is an a\times c matrix. For unit vectors u,v\in\R^4, define the orthogonal projections

These act on \R^a and \R^c, respectively. The next proposition recovers the scalar form (1.2) from the Schur threshold (4.1).

At u=v, the left side of (5.2) vanishes, so P(v)B(y)R(v)=0 for every y. We next study the identity as u varies near v.

6 The tangent obstruction

We finish the proof by taking a first-order limit in the norm identity at a point where the kernel projection varies smoothly. This produces a bilinear matrix whose entries must vanish by Lemma 2.1.

7 Removing a determinant factor

The polynomial p makes the positive-map construction explicit through a 4\times4 determinant. We now remove a factor that is unnecessary for its hyperbolicity cone. A larger block determinant shows directly that the quotient remains polynomial at singular X.

A A path-containment assertion

Theorem 53 (pp. 16–17) of version 1 of González Nevado’s [7] states a positive solution of the geometric Generalized Lax conjecture. Its proof invokes Theorem 52 (p. 16), which asserts containment of a fixed rigidly convex set throughout a smooth real-zero deformation from a product of normalized tangent linear factors to a multiple of the defining polynomial. The following example violates that intermediate assertion under its stated hypotheses.

A real polynomial g with g(0)=1 is real-zero if the roots of s\mapsto g(sx) are real for every real vector x. Its closed rigidly convex set is the closure of the component of \{g\ne0\} containing the origin. Fix a,b,\eta>0 and, in one variable, put

The zero set of g is the compact smooth set \{-b,a\}. The normalized tangent linear factors at these two points are 1+x/b and 1-x/a, whose product is g. Every H_t has value 1 at the origin and two distinct real roots,

Thus the path depends smoothly on t and consists of real-zero polynomials with compact smooth zero sets. It even admits the smooth monic real symmetric determinantal representation

Both endpoints equal g, so the final polynomial is g times the constant cofactor 1, as allowed by the stated hypotheses. Nevertheless the rigidly convex set of H_t is

which is strictly smaller than [-b,a] whenever 0<t<1. Consequently it does not contain the rigidly convex set of g.

The parameters a,b>0 describe the open family of normalized real quadratics with negative leading coefficient, so the failure is not confined to a symmetric or multiple-root example. The parameter \eta may be arbitrarily small. The cited Theorem 52 states no restriction to two or more variables, no lower bound on pencil size, and no requirement that the endpoint cofactor be nonconstant.

This example refutes the all-intermediate-times conclusion of that theorem. Its endpoints coincide, so it does not refute an endpoint-only statement. The nonspectrahedrality result of this paper follows from the construction and obstruction proved above.