咨询二项式$x^n+a$与三项式$x^r+bx^s+c$的结式相关研究论文

Hey there! Great question about resultants between binomials like $x^n + a$ and trinomials such as $x^r + bx^s + c$—this is a niche but super interesting corner of algebraic number theory and polynomial factorization. Let me share some resources and special case research I know about:

• Finite field-focused research: Since you referenced Richard Swan’s 1962 paper Factorization of Polynomials over Finite Fields (with its well-known binomial-binomial resultant lemma), there are direct follow-ups that tackle binomial-trinomial resultants in this context. Gary L. Mullen and Daniel Panario’s 1992 work Trinomials over Finite Fields dives into resultant-based criteria for irreducibility, which inherently involves calculating resultants between trinomials and binomials to check for common factors. This is a solid starting point if you’re working over finite fields.
• Lacunary polynomial resultants: For more general algebraic settings, look into research on lacunary polynomials (polynomials with few non-zero terms, which includes binomials and trinomials). Bernard Dwork’s p-adic analysis work includes studies on resultants of lacunary pairs, with specific cases covering binomial-trinomial combinations. While his work leans into p-adic theory, it provides a formal framework for computing these resultants.
• Special exponent cases: When dealing with specific exponent relationships (like $s = r/2$ for symmetric trinomials $x^{2k} + bx^k + c$, or small values of $n, r, s$), there are targeted papers and conference proceedings. For example, research on cyclotomic resultants by John H. Loxton and Andrew Granville includes simplified formulas for binomial-trinomial resultants where exponents have cyclotomic connections.

As you noted, binomial-binomial resultants are thoroughly covered (Swan’s 1962 Lemma 3 is indeed the standard reference here), but binomial-trinomial resultants lack a universal standardized formula—so focusing on special cases (finite fields, symmetric trinomials, constrained exponents) is the most productive path for existing research.

内容的提问来源于stack exchange，提问作者Gary McGuire