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In this paper, the Hamilton-Jacobi equation (HJE) with coefficients consisting of rational functions is considered, and its solutions with algebraic gradients are characterized in terms of commutative algebra. It is shown that there exists a solution with an algebraic gradient if and only if an involutive maximal ideal containing the Hamiltonian exists in a polynomial ring over the rational function field. If such an ideal is found, the gradient of the solution is defined implicitly by a set of algebraic equations. Then, the gradient is determined by solving the set of algebraic equations pointwise without storing the solution over a domain in the state space. Thus, the so-called curse of dimensionality can be removed when a solution to the HJE with an algebraic gradient exists. New classes of explicit solutions for a nonlinear optimal regulator problem are given as applications of the present approach.