Rational interpolants and solutions of dispersionless Hirota system
Andriy Panasyuk
TL;DR
This work constructs explicit rational solutions of the dispersionless Hirota system, whose solutions parametrize Veronese webs in $n$ dimensions. By representing the solution as a homogeneous rational function $f(x)=\frac{p_k(x)}{q_l(x)}$ and linking it to a Cauchy interpolant $F(λ,x)$ with nodes $λ_i$, the authors derive $f$ from $F$ via $f(x)=\frac{P_k(x)}{Q_l(x)}$, ensuring the associated one-form $α^λ$ Frobenius-integrable and thus yielding a Veronese web. The main result shows that nonflat Veronese webs arise when both $k$ and $l$ are positive, with flatness occurring for $(k,l)=(0,\cdot)$ or $(\cdot,0)$; explicit low-dimensional examples (3D–5D) illustrate the construction and the homogeneity properties $\deg p_k = \deg q_l+1$ and zero-sum coefficients. The paper also discusses leaves-restriction, potential deformations, and generalizations to Padé-type interpolation for deformed Hirota systems, highlighting broader geometric and analytic implications for Veronese webs.
Abstract
The aim of this paper is to construct a class of explicit nontrivial rational solutions of the dispersionless Hirota system of PDEs. All the solutions in this class are of homogeneity degree 1 and are quotients of homogeneous polynomials. It is well-known that the solutions of the Hirota dispersionless systems describe Veronese webs. By nontriviality of the solutions it is meant that the corresponding Veronese webs are nonflat at generic points.