Separant of an Arbitrary Polynomial

How to Find (Compute) a Separant

“…Corollary 3 in [2] shows that the constant σ f defies improvement under the hypotheses of Theorem 1.…”

“…In [2], a separant σ f of an arbitrary (unitary) polynomial f over a valued field F = F, R was defined to be an element of a value group Γ R 0 for any algebraically closed extension F 0 = F, R F. Namely, let f = i<n (x − α i ) be a factorization of f into linear factors over F 0 . For i < n, a local (in α i ) Krasner constant κ f,α i is defined thus:…”

“…In a series of papers [2][3][4][5][6], separants of polynomials were found for some important (interesting) classes of polynomials. If we use the notion of a constructive model [7] or, which is equivalent, of a computable model [7,8], then it is not hard to see that in the case where F 0 is a constructive field, the definition of a separant σ f itself shows a way of computing the separant.…”

“…Then ϕf k = f k since k f,ϕ(α i ) = k f,α i for any i < n. 2 We call f k , k < n, homogeneous factors of f and refer to the relation f = k≤deg f f k k as a canonical factorization. PROPOSITION 1.…”