Christol’s theorem [7] characterizes algebraic formal power series over finite fields in terms of automatic sequences, establishing a fundamental link between algebraicity and computability. A refinement of this result by Bell and the author [2] provides an algebraic characterization of formal power series whose support is a sparse automatic set, showing that sparseness can be characterized in terms of certain key transformations such as the Frobenius map, multiplicative scaling, and power transformations. In this paper, we extend these results to the multivariate setting, building on Salon’s [16] generalization of Christol’s theorem. By using a characterization of sparse regular languages, we look at algebraic multivariate formal power series whose supports are sparse subsets of \(\mathbb {N}^d\) and characterize them in terms of above-mentioned transformations, providing a structural understanding of the interplay between algebraicity and sparseness in multiple variables.