The number of real eigenvectors of a real polynomial
  I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, i prove that t is greater or equal than 2c+1, if d is odd and t is greater or equal than max(3,2c+1), is d is even, where c is the number of ovals of the locus of zeros of f; for binary forms, i prove that t is greater or equal than the number of real roots of f. Moreover, the above inequalities are the best possible for binary forms of any degree and for ternary cubic and quartic forms. 
Mauro Maccioni
math.AG