This paper establishes a constructive differential-algebraic framework for obtaining explicit analytic solutions to a broad class of nonlinear multivariate anti-difference equations. We define the nonlinear anti-difference algebraic closure KNLADE, a differentially closed field extension constructed via a recursive adjunction process that includes solutions of linearized anti-difference equations, multi-index radical extensions, roots of unity, and a predefined set of nonlinear special functions. Within this closure, we prove that solutions of nonlinear multivariate anti-difference equations with analytic kernels admit a unified representation of the form u(x) = u0(x) + M−1m=0(Φm(c,x))1/pmωkmpmψm(x)(Theorem 4.1). The framework rigorously addresses the inherent multidimensional and infinite dimensional challenges of anti-difference equations. We provide constructive proofs, derive explicit combinatorial expressions for the nonlinear correction coefficients, and establish convergence criteria for the iterative construction of nonlinear basis functions. Building on exterior summation theory, we demonstrate how multiple summation concepts are unified in a coordinate-free framework. Detailed algorithms with complexity analysis are presented, including stability guarantees and adaptive precision control. A rigorous verification framework with certified error bounds is established, employing interval arithmetic and cross-validation with high-order numerical methods. This work demonstrates that, while closed-form solutions in elementary functions are impossible for many nonlinear antidifference equations, explicit analytic solutions exist within the suitably extended and constructively defined nonlinear anti-difference algebraic closure KNLADE.
shifa liu (Wed,) studied this question.