Universal formulas for Chern classes of symmetric powers
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For each fixed k, can the elementary-basis coefficients of c_k(Symᵈ(Cⁿ)) be expressed as universal rational polynomials in T_r(n,d)=binom(d+n−1,n+r−1), 1≤r≤k?
Why it matters and what counts as progress
Why it matters: Universal dependence would replace isolated degree and Chern-class formulas by one framework across n and d.
What counts as progress: A proof of the polynomial formula for every k, explicit coefficient formulas in new cases, or a counterexample to the stated uniform dependence.
How the problem developed
- 1 Sept 2025 · Partial result [1]
László M. Fehér and Ádám Juhász Established polynomiality results for coefficients related to Chern classes of symmetric powers, providing a nearby formula framework.
- 24 May 2026 · Reported claim [2]
Gergely Bérczi and László M. Fehér Reported that for each fixed k, the elementary-basis coefficients of c_k(Sym^d(C^n)) are universal rational polynomials in the specified binomial terms T_r for r≤k.
AI contribution
Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics
Bérczi and Fehér report that, for every k, the elementary-basis coefficients of the kth Chern class of Sym^d(C^n) are universal rational polynomials in T_r(n,d)=binom(d+n−1,n+r−1), for 1≤r≤k.
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