Weyl and Lidskĭı inequalities for general hyperbolic polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidskĭı inequalities. We give an elementary proof of the latter for hyperbolic polynomials. This proof is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory. Notations: We deal with n-uplets of real numbers, indexed by {1, . . . , n}. The set I r of parts I ⊂ {1, . . . , n} of cardinality r has itself cardinality ( n r ) = n! r!(n− r)! . We order I r in the following natural way. Each element I ∈ I r consists in indices (1 ≤) i1 < · · · < ir (≤ n). We say that I ≺ J if i1 ≤ j1, . . . , ir ≤ jr, and we say that I is lower than J , or that J is higher than I. Given two elements I,K in I r , we form the segment [I,K] of elements J ∈ I r such that I ≺ J ≺ K ; it is void unless I ≺ K. Likewise, (I,K], [I,K) and (I,K) are the segments where we require additionally that either J 6= I or J 6= K or both. Although ≺ is not a total order, the class I r has a lowest element minr = {1, . . . , r} and a highest one maxr = {n− r+1, . . . , n}. Unless ambiguity, we simply write min and max for minr and maxr. The “open” segment (min,max) also has a lowest element supmin := {1, . . . , r − 1, r + 1} and a highest one submax := {n− r, n− r + 2, . . . , n}. We define the order-reversing involution π over {1, . . . , n} by πj := n+ 1− j ; π acts likewise on I r by π{i1, . . . , ir} := {n− ir + 1, . . . , n− i1 + 1}. ∗UMPA (UMR 5669 CNRS), ENS de Lyon, 46, allée d’Italie, F–69364 Lyon, cedex 07, FRANCE.