arXiv subjects
Publications and source records attributed to N. Perminov.
In this paper, we obtain an explicit expression for the resultant of n quadratic algebraic equations dS/dx_1 = 0,..., dS/dx_n = 0, where S is a cubic polynomial in n variables, symmetric under permutations of its arguments. Application of this result to the study of Finslerian spaces is discussed.
A homogeneous polynomial S(x_1, ..., x_n) of degree r in n variables posesses a discriminant D_{n|r}(S), which vanishes if and only if the system of equations dS/dx_i = 0 has non-trivial solutions. We give an explicit formula for discriminants of symmetric (under permutations of x_1, ..., x_n) homogeneous polynomials of degree r in n >= r variables. This formula is division free and quite effective from the computational point of view: symbolic computer calculations with the help of this formula take seconds even for n ~ 20. We work out in detail the cases r = 2,3,4 which will be probably important in applications. We also consider the case of completely antisymmetric polynomials.