The regularized second-kind Abelian integral on a non-hyperelliptic (n,s)-curve — the plane canonical form yn+xs+(moduli terms)=0y^n+x^s+(\text{moduli terms})=0 whose nn sheets all meet at the single point at infinity — carries a vector of free constants c(n,s)(λ)c_{(n,s)}(\lambda), uniquely fixed by the requirement that all relations among the Abelian functions on the curve's Jacobian be consistent. Bernatska and Leykin (SIGMA 14 (2018) 074) computed these regularization constants for the (3,4), (3,5), (3,7) and (4,5) curves and verified, for (3,4) and (3,5), the conjecture of Bukhshtaber and Leykin (2005) that the primitive function — the exponential of the regularized second-kind integral — coincides, up to a rational-number factor, with the first non-vanishing derivative of the multivariate sigma function on the Abel image. The general problem remains open: no proof valid for all coprime (n,s)(n,s), no closed-form c(n,s)(λ)c_{(n,s)}(\lambda), and no uniform addition laws exist; the authors' own 2025 survey still records all three gaps. The problem asks for the theory uniform in (n,s)(n,s): the general identification proof, the general constant reproducing the known cases, and the addition laws on the Jacobian that the polylinear sigma identities then yield.

Background

On a compact Riemann surface XX of genus gg, Abelian integrals are the higher-genus analogue of elliptic integrals: integrals of everywhere-holomorphic differentials are of the first kind, and integrals of meromorphic differentials with vanishing residues are of the second kind. Integrals of the first kind define Abel's map A\mathcal{A}, which sends a divisor (a formal sum of curve points) into the Jacobian variety Jac⁡(X)=Cg/Λ\operatorname{Jac}(X)=\mathbb{C}^g/\Lambda, a complex torus on which all multiply periodic functions of the theory live.

This problem concerns the so-called (n,s)-curves: for coprime integers n<sn<s, f(n,s)(x,y)=yn+xs+∑i=0s−2∑j=0n−2λns−in−js xiyj=0,f_{(n,s)}(x,y)=y^n+x^s+\sum_{i=0}^{s-2}\sum_{j=0}^{n-2}\lambda_{ns-in-js}\,x^iy^j=0, with λk∈C\lambda_k\in\mathbb{C} and λk=0\lambda_k=0 for k≤0k\le 0. A non-degenerate (n,s)-curve — one of maximal genus g=(n−1)(s−1)/2g=(n-1)(s-1)/2 in this family — has a single point at infinity through which all nn sheets of the curve pass; this point is a Weierstrass point and the natural basepoint of Abel's map. These curves directly generalize Weierstrass's canonical form of elliptic curves and serve as canonical representatives of birational equivalence classes of plane algebraic curves with the corresponding singularity structure at infinity. The monoid of non-negative combinations ⟨n,s⟩\langle n,s\rangle is the Weierstrass semigroup at infinity; its complement in the non-negative integers, of cardinality gg, is the gap sequence, which orders the coordinates u1,…,ugu_1,\dots,u_g of Abel's map by Sato weight (wgt⁡x=n\operatorname{wgt}x=n, wgt⁡y=s\operatorname{wgt}y=s, and the weights of the uiu_i are read off from the gaps).

On the Jacobian of an (n,s)-curve one has the multivariate sigma function σ(u)\sigma(u), u∈Cgu\in\mathbb{C}^g: the unique solution of a system of linear heat equations, posed in a specially weighted (so-called nonholonomic) coordinate frame on the Jacobian, whose initial condition is the Schur polynomial of the gap sequence (the Schur–Weierstrass polynomial). It admits a convergent expansion into Sato-weight-homogeneous polynomials in the curve parameters λ\lambda and powers of uu. Its logarithmic derivatives produce the Kleinian (multiply periodic) ℘\wp-functions, ℘i,j(u)=−∂ui∂ujlog⁡σ(u)\wp_{i,j}(u)=-\partial_{u_i}\partial_{u_j}\log\sigma(u), which generate the field of Abelian functions on the Jacobian and satisfy systems of algebraic relations analogous to the Weierstrass elliptic relation ℘′2=4℘3−g2℘−g3\wp'^2=4\wp^3-g_2\wp-g_3.

To each first-kind differential duidu_i of the special cohomology basis (the basis of first- and second-kind differentials adapted to the point at infinity) is associated a second-kind differential dridr_i, holomorphic on the curve punctured at infinity with a pole there. Because the basepoint of Abel's map is exactly this pole, the second-kind integral requires regularization: parameterizing the curve by a local parameter ξ\xi at infinity, one defines the regularized second-kind integral r(ξ)=c+∫0ξdrr(\xi)=c+\int_0^{\xi}dr, and the free constant vector c=c(n,s)(λ)c=c_{(n,s)}(\lambda) — the regularization constant, a uniquely defined free term in the expansion of the second-kind integral over the local parameter near infinity — must be chosen so that all relations among Abelian functions constructed from these integrals are consistent. In the elliptic case this is Weierstrass's regularization of the zeta function. Bernatska and Leykin (2018) proved uniqueness of the constant ('consistent with the structure of the Abelian function field') and computed, for example, c(3,4)(λ)=(0,−λ2/3,−λ5/6)tc_{(3,4)}(\lambda)=(0,-\lambda_2/3,-\lambda_5/6)^t, together with the analogous vectors for (3,5), (3,7) and (4,5); in the hyperelliptic case all components vanish.

The regularized integral defines the primitive function ψ(ξ)=exp⁡(−∫ξB du)\psi(\xi)=\exp(-\int^{\xi}B\,\mathrm{d}u), where BB is the vector of regularized second-kind integrals; ψ\psi is an entire function of the local parameter with a zero of order gg at ξ=0\xi=0, of the form ψ(ξ)=ξgexp⁡T(ξ,λ)\psi(\xi)=\xi^g\exp T(\xi,\lambda) with TT holomorphic. It is the key ingredient of polylinear identities between sigma functions, such as the bilinear identity σ(u−A(ξ))σ(u+A(ξ))/(ψ2(ξ)σ2(u))=φ2g(x(ξ),y(ξ);u)\sigma(u-\mathcal{A}(\xi))\sigma(u+\mathcal{A}(\xi))/(\psi^2(\xi)\sigma^2(u))=\varphi_{2g}(x(\xi),y(\xi);u) with φ2g\varphi_{2g} a rational function of Sato weight 2g2g on the curve whose coefficients are polynomials in the ℘\wp-functions; expanding such identities in ξ\xi yields hierarchies of relations among ℘\wp-functions, and the trilinear version yields addition laws on the Jacobian — the higher-genus analogue of the Weierstrass elliptic addition theorem. The central structural conjecture, stated by Bukhshtaber and Leykin (2005) for general (n,s)-curves (their Theorem 9) but proven by them only in the hyperelliptic case, and proven by Bernatska–Leykin (2018) for the (3,4) and (3,5) curves, is that ψ(ξ)\psi(\xi) coincides, up to a rational-number factor, with the first non-vanishing derivative of σ\sigma at u=A(ξ)u=\mathcal{A}(\xi) — equivalently, the sigma-derivative whose differential operator has Sato weight −wgt⁡σ−g-\operatorname{wgt}\sigma-g, all derivatives of lower Sato weight vanishing along the Abel image. This identification holds only for the correct regularization constant, so the closed form of c(n,s)(λ)c_{(n,s)}(\lambda) and the proof of the identification are two faces of one problem.

What is missing is the general theory uniform in (n,s)(n,s). The two available computation methods are case-by-case and do not scale: Method A (matching the sigma-series up to Sato weight 2g−12g-1) requires a number of terms that grows exponentially with the genus, and Method B (comparing the relations among Abelian functions produced by the bilinear identity with independently derived ones, via the residue theorem and Klein's fundamental 2-form of the second kind, constructed for the wider class of CabC_{ab} curves (plane curves ya=Pb(x)y^{a}=P_{b}(x), which contain the (n,s)-curves) by Suzuki) required 2, 7, 16 and 18 independent relations for (3,4), (3,5), (3,7) and (4,5) respectively. Adjacent progress has not closed the gap: Nakayashiki expressed the (n,s)-sigma function through algebraic integrals, proving that the leading term of its series at the origin is the Schur function of the gap sequence; Komeda–Matsutani–Previato extended the sigma function and the second-kind differentials normalized by the Weierstrass gap structure (W-normalized) to general Weierstrass curves; and Bernatska–Leykin (2023) solved the Jacobi inversion problem on non-hyperelliptic (n,s)-curves by a residue method that bypasses the regularization constant altogether. The 2025 survey of Abelian function fields still records the constants as known only in the four cases, presents the general polylinear identity only as an unproven generalization of the hyperelliptic result, and calls the polylinear-operator program open.

Problem Statement

Establish the regularization theory of second-kind integrals on non-degenerate non-hyperelliptic (n,s)-curves uniformly in the coprime pair (n,s)(n,s) with 3≤n<s3\le n<s. Concretely, prove for all coprime pairs (n,s)(n,s) with 3≤n<s3\le n<s simultaneously:

(1) the primitive-function identification: ψ(ξ)\psi(\xi), defined from the regularized second-kind integral with basepoint at the point at infinity, coincides up to multiplication by a rational number with the first non-vanishing derivative (in Sato-weight order) of the multivariate sigma function at u=A(ξ)u=\mathcal{A}(\xi) — currently proven only for hyperelliptic curves and for the (3,4)(3,4)- and (3,5)(3,5)-curves;

(2) a general closed-form expression for the regularization constant c(n,s)(λ)c_{(n,s)}(\lambda) (equivalently, a uniform explicit determination of it valid for arbitrary (n,s)(n,s) with 3≤n<s3\le n<s), together with a proof that it is the unique constant consistent with the field of Abelian functions; the formula must reproduce the published values for (3,4)(3,4), (3,5)(3,5), (3,7)(3,7), (4,5)(4,5) and the vanishing constant in the hyperelliptic case;

(3) the polylinear (bilinear and trilinear) sigma identities and the induced addition laws for basis Kleinian ℘\wp-functions on the Jacobian of an arbitrary non-hyperelliptic (n,s)-curve.

An answer must be valid for all coprime pairs (n,s)(n,s) with 3≤n<s3\le n<s at once: verifying yet another fixed pair by case-by-case computation — sigma-series matching to Sato weight 2g−12g-1, or assembling finitely many independent relations among Abelian functions — constitutes progress toward the problem but does not by itself complete it.

References

  1. J. Bernatska, D. Leykin, On Regularization of Second Kind Integrals, SIGMA 14 (2018), 074, 28 pages; DOI 10.3842/SIGMA.2018.074; arXiv:1709.10167; https://arxiv.org/abs/1709.10167
  2. V. M. Buchstaber (also transliterated V. M. Bukhshtaber), D. V. Leykin (also transliterated D. V. Leikin), Addition laws on Jacobian varieties of plane algebraic curves, Proc. Steklov Inst. Math. 251 (2005), 49–120; Russian original: Tr. Mat. Inst. Steklova 251 (2005), 54–126; https://www.mathnet.ru/eng/tm45
  3. J. Bernatska, D. Leykin, Solution of the Jacobi inversion problem on non-hyperelliptic curves, Letters in Mathematical Physics 113 (2023), no. 5, Article 110; DOI 10.1007/s11005-023-01726-3; arXiv:2212.14492; https://arxiv.org/abs/2212.14492
  4. J. Bernatska, Abelian Function Fields on Jacobian Varieties, Axioms 14 (2025), no. 2, art. 90; DOI 10.3390/axioms14020090; arXiv:2412.05455; https://arxiv.org/abs/2412.05455
  5. J. Komeda, S. Matsutani, E. Previato, Algebraic Construction of the Sigma Function for General Weierstrass Curves, Mathematics 10 (2022), no. 16, art. 3010; DOI 10.3390/math10163010; arXiv:2207.02690; https://arxiv.org/abs/2207.02690
  6. A. Nakayashiki, On Algebraic Expressions of Sigma Functions for (n,s) Curves, Asian Journal of Mathematics 14 (2010), 175–212; DOI 10.4310/ajm.2010.v14.n2.a2; arXiv:0803.2083; https://arxiv.org/abs/0803.2083
  7. J. Suzuki, Klein's Fundamental 2-Form of Second Kind for the CabC_{ab} Curves, SIGMA 13 (2017), 017, 13 pages; DOI 10.3842/SIGMA.2017.017; arXiv:1701.00931; https://arxiv.org/abs/1701.00931
  8. J. Bernatska, Y. Kopeliovich, Addition via reduction algorithm on trigonal curves, arXiv:2006.11838; https://arxiv.org/abs/2006.11838