Unified Analytic Solution of Integral Equations in Integral Algebraic Closure

This paper establishes a rigorous integral algebraic framework for solving integral equations of various types, extending the differential algebraic closure approach developed for polynomial equations. We prove that solutions to integral equations can be analytically expressed within an integral algebraic closure Kint, which extends the coefficient field with integral operators and their evaluations. For Hammerstein-type equations with polynomial nonlinearities of degree n, the solution formula takes the unified form: N ϕk(x) = ϕ(n−1)(x) + m=1 Φm(y)1/nωm(k−1) n ψm(x), 0 ≤k ≤n−1, where ϕ(n−1) is the integral critical function, y = (y(0),...,y(n−2)) are integral critical values, Φm ∈ Q(a)[y] are explicit polynomials with combinatorial correction terms, ωn = e2πi/n, and {ψm} form a complete orthonormal basis. We provide constructive proofs with enhanced mathematical rigor, derive combinatorial expressions for correction coefficients γ(n) m with detailed verification, and present detailed numerical algorithms with comprehensive parameter selection strategies. Extensive validation demonstrates machine-precision accuracy for various integral equation types.