A theorem on the escape from the space of hyperbolic polynomials
A hyperbolic polynomial \(P\) is a real polynomial in one variable whose roots are all real. We answer to the following question: For which integers \(s<\deg P\) is it possible to find a polynomial \(Q\) of degree \(s\) such that \(P+Q\) remains hyperbolic? More precisely we will use Arnol'd's terminology: Let \(s\) be an integer such that \(0\leq s<\deg P\), we will say that the monic hyperbolic polynomial \(P\) is globally \(s\)-maximal (respectively globally \(s\)- minimal) if \(P+c_ 0x^ s+\cdots+c_ s\) with \(c_ 0>0\) (respectively \(c_ 0<0)\) is never hyperbolic. The monic hyperbolic polynomial \(P\) is locally \(s\)-maximal (respectively locally \(s\)-minimal) if \(P+c_ 0x^ s+\cdots+c_ s\) with \(c_ 0>0\) (respectively \(c_ 0<0)\) is not hyperbolic for \(c_ 0,\dots,c_ s\) small enough. We prove the equivalence between the local property and the global property and that these properties depend only on the multiplicity vector of \(P\), which we define as follows: If \(P\) is a monic hyperbolic polynomial, then it may be written uniquely as \(P=(X-x_ 1)^{m_ 1}\dots(X-x_ k)^{m_ k}\) with \(x_ 1>x_ 2>\cdots>x_ k\) and the \(k\)-tuple \((m_ 1,\dots,m_ k)\) is called the multiplicity vector of \(P\). To any monic hyperbolic polynomial \(P\) we may associate an integer \(s_ P\) which depends only on the multiplicity-vector of \(P\) as follows: (i) if \(P\) has only simple roots, \(s_ P=-1\). (ii) if not, let \(s_ P=l+\sum_{i/m_ i\geq 2}(m_ i-2)\), where \(l\) is the number of odd sequences of consecutive 1 in the multiplicity vector, considering only sequences which are between two multiplicities greater than or equal to 2. The hyperbolic polynomial \(P\) is called even when its multiplicity vector begins with an even sequence of consecutive 1, and odd otherwise. With these notations we have the following results: Theorem: Let \(P\) be a monic hyperbolic polynomial. a) \(P\) is locally \(s\)-maximal (respectively minimal) if and only if \(P\) is globally \(s\)-maximal (respectively minimal). b) If \(0\leq s<s_ P\), then \(P\) is \(s\)-maximal and \(s\)-minimal. c) If \(s=s_ P\) and \(P\) is even, then \(P\) is \(s\)-maximal but not \(s\)- minimal. d) If \(s=s_ P\) and \(P\) is odd, then \(P\) is \(s\)-minimal but not \(s\)- maximal. e) If \(s>s_ P\), then \(P\) is neither \(s\)-maximal nor \(s\)-minimal. Another proof of a) can be deduced from works of Arnol'd, Givental' and Kostov.
scientific article; zbMATH DE number 126406