A sequence is difference algebraic (or D-algebraic) if finitely many shifts of its general term satisfy a polynomial relationship; that is, they are the coordinates of a generic point on an affine hypersurface. The corresponding equations are called algebraic difference equations (ADE). We show that subsequences of D-algebraic sequences, indexed by arithmetic progressions, satisfy ADEs of the same orders as the original sequences. Additionally, we provide algorithms for operations with D-algebraic sequences and discuss the difference-algebraic nature of holonomic and $C^2$-finite sequences.

翻译：若序列的通项有限次平移后满足多项式关系，即它们是仿射超曲面上一般点的坐标，则称该序列为差分代数（或 D-代数）序列。相应的方程称为代数差分方程（ADE）。我们证明了 D-代数序列中由算术级数索引的子序列满足与原始序列同阶的 ADE。此外，我们提供了 D-代数序列的运算算法，并讨论了全纯序列与 $C^2$-有限序列的差分代数性质。