SOS锥边界上的严格正多项式

Facial reduction for exact polynomial sum of squares decompositions

We study the problem of decomposing a non-negative polynomial as an exact sum of squares (SOS) in the case where the associated semidefinite program is feasible but not strictly feasible (for example if the polynomial has real zeros). Computing symbolically roots of the original polynomial and applying facial reduction techniques, we can solve the problem algebraically or restrict to a subspace where the problem becomes strictly feasible and a numerical approximation can be rounded to an exact solution. As an application, we study the problem of determining when can a rational polynomial that …

Symmetry Adapted Gram Spectrahedra

This paper explores the geometric structure of the spectrahedral cone, called the symmetry adapted PSD cone, and the symmetry adapted Gram spectrahedron of a symmetric polynomial. In particular, we determine the dimension of the symmetry adapted PSD cone, describe its extreme rays, and discuss the structure of its matrix representations. We also consider the symmetry adapted Gram spectrahedra for specific families of symmetric polynomials including binary symmetric polynomials, quadratics, and ternary quartics and sextics which give us further insight into these symmetric SOS polynomials. Fina…

Blenders

A blender is a closed convex cone of real homogeneous polynomials that is also closed under linear changes of variable. Non-trivial blenders only occur in even degree. Examples include the cones of psd forms, sos forms, convex forms and sums of $2u$-th powers of forms of degree $v$. We present some general properties of blenders and analyze the extremal elements of some specific blenders.

Geometry of totally real Galois fields of degree 4

We will consider a totally real Galois field $K$ of degree 4 as the linear coordinate space $\mathbb{Q}^4\subset\mathbb{R}^4$. An element $k\in K$ is called strictly positive, if all its conjugates are positive. The set of strictly positive elements is a convex cone in $K$. The convex hull of strictly positive integral elements is a convex subset of this cone and its boundary $Γ$ is an infinite union of 3-dimensional polyhedrons. The group $U$ of strictly positive units acts on $Γ$: the action of a strictly positive unit permutes polyhedrons. Fundamental domains of this action are the object o…

Nonnegative Polynomials and Sums of Squares

In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of nonnegative polynomials is that polynomials of degree $d$ satisfy certain linear relations, known as the Cayley-Bacharach relations, which are not satisfied by polynomials of full degree 2d. For any nonnegative polynomial that is not a sum of squares we can write down a linear inequality coming from a Cayley-Bacharach relation that certifies this fact. We a…

严格双曲多项式的广义Lax猜想

The Generalized Lax Conjecture for Strictly Hyperbolic Polynomials

该论文报告了AI系统证明严格双曲多项式的双曲锥为谱面体，从而在一般情形下确立了广义Lax猜想，但完整猜想仍开放。

Intermediate Cones between the Cones of Positive Semidefinite Forms and Sums of Squares

The cone $\mathcal{P}_{n+1,2d}$ ($n,d\in\mathbb{N}$) of all positive semidefinite (PSD) real forms in $n+1$ variables of degree $2d$ contains the subcone $Σ_{n+1,2d}$ of those that are representable as finite sums of squares (SOS) of real forms of half degree $d$. In 1888, Hilbert proved that these cones coincide exactly in the Hilbert cases $(n+1,2d)$ with $n+1=2$ or $2d=2$ or $(n+1,2d)=(3,4)$. To establish the strict inclusion $Σ_{n+1,2d}\subsetneq\mathcal{P}_{n+1,2d}$ in any non-Hilbert case, one can show that verifying the assertion in the basic non-Hilbert cases $(4,4)$ and $(3,6)$ suffic…

The Moment Problem for Continuous Positive Semidefinite Linear functionals

Let $τ$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ top…

关于有界比率的注记

A note on bounded ratios

该数学注记证明半代数集上的有界比率集合为有理多面体凸锥，给出洛伦兹多项式有界比率相关猜想的反例，关联7顶点割锥的非超度量团-网面。

多项式优化中的精确矩表示

Exact Moment Representation in Polynomial Optimization

针对多项式优化中实半代数集上多项式下确界问题，研究矩矩阵（MoM）层次结构的精确性，揭示其与扩展实根的平方和（SoS）层次结构的对偶性，证明平坦截断与支撑集零维的等价性，并通过边界黑塞条件等验证精确性。

Hyperbolicity cones and imaginary projections

Recently, the authors and de Wolff introduced the imaginary projection of a polynomial $f\in\mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. In this article, we study the relation between the imaginary projections and hyperbolicity cones, where the latter ones are only defined …

秩至少为5的对称锥上的协正锥和完全正锥

Copositive and completely positive cones over symmetric cones of rank at least 5

研究秩至少为5的对称锥上的协正锥和完全正锥，证明它们不是谱面阴影，引入Horn变换生成暴露射线，并揭示平方和内部逼近层次在正半定锥上不是渐近精确的。

Tangent cones and regularity of real hypersurfaces

We characterize embedded $\C^1$ hypersurfaces of $\R^n$ as the only locally closed sets with continuously varying flat tangent cones whose measure-theoretic-multiplicity is at most $m<3/2$. It follows then that any (topological) hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is $\C^1$. In the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface. In particular, any convex real analytic hypersurface $X\subset\R^n$ is $\C^1$. Furthermore, if $X$ is real algebraic, strictly convex, and …