论文标题
双曲线多项式和刚性模量订单
Hyperbolic polynomials and rigid moduli orders
论文作者
论文摘要
双曲线多项式（HP）是一个真正的单变量多项式，所有根源是真实的。根据Descartes的标志规则，所有系数的HP规则都具有$ C $的正面和$ p $ p $ pug的负数，其中$ c $和$ p $是其系数的符号更改和符号保存的数量。我们认为具有不同根部模量的HP。我们问一个问题。实际正半线上的正根完全决定了多项式系数的符号。当至少有一个正质和至少一个负根时，当负根的模量与正根的模量（因此，根的一半或一半是阳性）时，可能会准确。在这种情况下，HP系数的符号为$（+，+， - ， - ， - ，+，+， - ， - ， - \ ldots）$或$（+， - - ， - ， - ，，+，+， - ， - ， - ， - +，\ ldots）$。
A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where $c$ and $p$ are the numbers of sign changes and sign preservations in the sequence of its coefficients. We consider HPs with distinct moduli of the roots. We ask the question when the order of the moduli of the negative roots w.r.t. the positive roots on the real positive half-line completely determines the signs of the coefficients of the polynomial. When there is at least one positive and at least one negative root this is possible exactly when the moduli of the negative roots interlace with the positive roots (hence half or about half of the roots are positive). In this case the signs of the coefficients of the HP are either $(+,+,-,-,+,+,-,-,\ldots )$ or $(+,-,-,+,+,-,-,+,\ldots )$.
