关于三次格上$3 \times 3$行列式相合问题中的初值问题
On Initial Data in the Problem of Consistency on Cubic Lattices for $3 \times 3$ Determinants
摘要：
本文给出了立方格上$3 \times 3$行列式相容性定理的完整证明。考虑了在格点$\mathbb{Z}^2$上形成基本$3 \times 3$正方形的点处标量场值的所有$3 \times 3$矩阵的行列式为零的条件下，定义的$\mathbb{Z}^2$上的离散非线性方程，给出了初始数据上一般位置的一些显式具体条件;并且对于满足这些一般位置的具体条件的任意初始数据，证明了所考虑的由$3 \times 3$行列式定义的$\mathbb{Z}^2$上的离散非线性方程组在立方格上的一致性定理（围绕立方体的一致性）.
英文摘要：
The paper is devoted to complete proofs of theorems on consistency on cubic lattices for $3 \times 3$ determinants. The discrete nonlinear equations on $\mathbb{Z}^2$ defined by the condition that the determinants of all $3 \times 3$ matrices of values of the scalar field at the points of the lattice $\mathbb{Z}^2$ that form elementary $3 \times 3$ squares vanish are considered; some explicit concrete conditions of general position on initial data are formulated; and for arbitrary initial data satisfying these concrete conditions of general position, theorems on consistency on cubic lattices (a consistency "around a cube") for the considered discrete nonlinear equations on $\mathbb{Z}^2$ defined by $3 \times 3$ determinants are proved.
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