Discriminant of an arbitrary polynomial composition

Determine the discriminant of the composition f(g(x)) for arbitrary monic polynomials f(x) and g(x), expressing it explicitly in terms of their coefficients and exponents.

Background

The paper studies discriminants of compositions over valued fields and proves an explicit formula when the inner polynomial is a trinomial of the form h(x)=xm+a xk+b, subject to the stated hypotheses. The authors present the broader problem for arbitrary monic inner and outer polynomials, which is not resolved by the trinomial formula.

The unresolved issue is to obtain a general coefficient-level discriminant formula for f(g(x)) without restricting g(x) to the trinomial family. Such a formula would extend the paper’s Theorem 1 and provide an efficient method for computing discriminants of more general composed polynomials.

References

To better understand this broader class of polynomials, particularly in higher degrees, a natural direction is to consider the following question. Let $f(x)$ and $g(x)$ be arbitrary monic polynomials. What is the discriminant of the composition $f(g(x))$ in terms of their coefficients and exponents?

— On The Index of Polynomial Compositions over Valued Fields  (2610.02111 - Jakhar et al., 1 Oct 2026) in Section 1, Introduction, immediately before Theorem 1 (page number unavailable)