A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article are families of hyperbolic polynomials which are determined through $k$ linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a \emph{hyperbolic slice}. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the \emph{local extreme points} of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most $k$ distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

翻译：一个$n$次实系数单变量多项式称为双曲的，若其所有$n$个根均位于实轴上。此类多项式在组合学与优化等不同应用中自然出现。本文聚焦于由系数上的$k$个线性条件所确定的双曲多项式族。对应于该双曲多项式族的系数构成一个半代数集，我们称之为双曲切片。本文从更详细的角度出发，研究了这些对象的几何性质。双曲多项式集根据实零点的重数自然分层，此分层亦诱导了双曲切片上的分层。我们主要关注双曲切片的局部极值点，即线性泛函的局部极值点，并证明这些极值点恰好对应于双曲切片中至多具有$k$个不同根的双曲多项式，且我们可证明在一般情况下，此类族的凸包是一个多面体。基于这些结果，我们将其应用于对称实簇与对称半代数集的研究中。在此，我们证明由可用初等对称多项式稀疏表示的对称多项式定义的集合，可在具有少量不同坐标的点上进行采样。这进而允许算法上的简化，例如验证此类多项式的非负性或由这些多项式定义的半代数集是否为空集。