The Four-Point Picard Theorem for Quaternionic Slice Regular Functions

Abstract

An entire slice regular function $f:\mathbb{H}\to \mathbb{H}$ can omit four prescribed quaternionic values only in the affine-dependent case. More precisely, four affinely independent omitted values force $f$ to be constant, while the converse follows from the plane-omission theorem of Bisi--Winkelmann. The proof passes to the real-symmetric stem function. For each omitted value a quadratic zero-divisor criterion gives a zero-free entire function $Q_j$, and the component normal to the affine span is governed by a square-discriminant identity. Finite-order data are excluded by Hadamard factorization and a rigidity argument on the real axis. In the general case, logarithmic Bloch--Ochiai places the $Q$-curve in a translated algebraic torus. The Laurent-square case reduces to the finite-order contradiction, and the nonsquare case is excluded by an even-ramification argument together with the level-one truncated Second Main Theorem of Noguchi--Winkelmann--Yamanoi.