The second discriminant of a univariate polynomial  

作　　者：Dongming Wang Jing Yang 

机构地区：[1]BDBC,LMIB and School of Mathematics and Systems Science,Beihang University,Beijing 100191,China [2]SMS,HCIC,Guangxi University for Nationalities,Nanning 530006,China [3]Centre National de la Recherche Scientifique,Paris 75794,France

出　　处：《Science China Mathematics》2021年第6期1157-1180,共24页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.61702025 and 11801101);the Special Fund for Guangxi Bagui Scholar Project;Guangxi Science and Technology Program(Grant No.2017AD23056);the Startup Foundation for Advanced Talents in Guangxi University for Nationalities(Grant No.2015MDQD018)。

摘　　要：We define the second discriminant D_(2)of a univariate polynomial f of degree greater than 2 as the product of the linear forms 2r_(k)-r_(i)-r_(j)for all triples of roots r_(i),r_(k),r_(j)of f with i<j and j≠k,k≠i.D_(2)vanishes if and only if f has at least one root which is equal to the average of two other roots.We show that D_(2)can be expressed as the resultant of f and a determinant formed with the derivatives of f,establishing a new relation between the roots and the coefficients of f.We prove several notable properties and present an application of D_(2).

关 键 词：DETERMINANT DISCRIMINANT polynomial ideal RESULTANT root configuration 

分 类 号：O174.14[理学—数学]