Results on hyperbolic polynomials and the cone: Gårding's theorem, the Choi–Lam form, the positive map, the slices and the obstruction, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• A nonspectrahedral hyperbolicity cone (OpenAI) — Theorem A. p(X,Z,y) = det((det X)Z - Phi_y(adj X)) is hyperbolic with respect to (I,I,0), and no linear matrix inequality of any size describes its closed hyperbolicity cone. Proved in the last lesson.
• Degree sixteen (OpenAI) — Corollary B. p/det X is a hyperbolic polynomial of degree 16 with the same cone. Proved in the third lesson.
• Gårding's theorem — Theorem 1.2. The open hyperbolicity cone is the component of {p != 0} containing e; p is hyperbolic with respect to each of its points; the open and closed cones are convex.
• Facts on symmetric matrices — Lemma 1.3. Square roots, the variational formula and continuity of the largest eigenvalue, congruence, the Schur complement, kernels, and rI >= C^T C exactly when r >= ||C||^2.
• Determinants of definite pencils — Proposition 1.4. If L(e) > 0, det L(x) is hyperbolic with respect to e and its closed cone is {L(x) >= 0}.
• Nonnegativity of the Choi–Lam form — Lemma 2.2. b(z,y) >= 0, with b(e1,e1) = 1.
• No dominated bilinear square (OpenAI) — Lemma 2.3. A bilinear form l with l(z,y)^2 <= b(z,y) for all z, y is zero.
• Not a sum of squares — Corollary 2.4. The Choi–Lam form is not a sum of squares of bilinear forms.
• The positive map Phi_y — Lemma 3.1. u^T Phi_y(vv^T) u = b(v_0 u' - u_0 v', y); hence Phi_y preserves positive semidefiniteness.
• Hyperbolicity of p (OpenAI) — Proposition 3.2. p is homogeneous of degree 20, p(e) = 1, hyperbolic with respect to e, and p = (det X)^4 det(Z - Phi_y(X^-1)) for invertible X.
• Two slices of the cone (OpenAI) — Proposition 3.3. For X > 0, (X,Z,y) lies in the cone exactly when Z >= Phi_y(X^-1); (X,Z,0) lies in it exactly when X, Z >= 0.
• A pencil gives a Schur-complement inequality (OpenAI) — Proposition 4.1. A pencil describing the cone yields unital positive maps D, E and a linear B with Z >= Phi_y(X^-1) exactly when E(Z) >= B(y)^T D(X)^-1 B(y).
• The norm identity (OpenAI) — Proposition 4.3. b(v_0 u' - u_0 v', y) = ||P(v) B(y) R(u)||^2 for unit u, v, with kernel projections P(v), R(u).
• An exact semidefinite lift (OpenAI) — Theorem C. The cone is the projection of a spectrahedron: there are A_1..A_23, G_1..G_307 in S^100 with x in the cone exactly when sum x_nu A_nu + sum v_j G_j >= 0 for some v. Proved in the last lesson.
• Angular Gram factorization (OpenAI) — Lemma 5.1. On the plane y = eta d(t) + xi k, Q(y) = F F^T for an explicit 3x4 matrix F linear in (eta, xi) and trigonometric of degree 3 in t.
• Spectrahedral slices (OpenAI) — Proposition 5.3. p(te - x) = det(tI_20 - L_t(x)) for x on the plane of angle t, so each angular slice of the cone is described by L_t >= 0.
• Degree-two matching (OpenAI) — Lemma 6.1. For S symmetric and z with (z1, z2) != 0 there is C(t) of trigonometric degree 2 with C C^T = Q(z) and <F_t(eta, xi), C(t)>_S = tr(S Q(Y_t(eta, xi), z)).
• The moment lift (OpenAI) — Proposition 6.2. (X, Z, y) lies in the cone exactly when some linear functional on E satisfies the output equations and H(Lambda) >= 0.

Reading order

Read the lessons in order. The second lesson (the Choi–Lam form) is elementary and can be read on its own; the first contains a complete proof of Gårding's theorem, used only for Corollary B. The last two lessons, on the exact lift, build on the first three; the fourth is needed there only for the comparison with Theorem A.