Abstract
We show that the ideal generated by particular finite linear combinations of shift operators in \( \mathbb{R} \times \mathbb{R} \) contains certain operators whose associated equations can be solved. Since these solutions include all solutions to the equations whose operators generated the ideal, they can then be used to solve a polygonal functional equation in \( \mathbb{R} \times \mathbb{R} \) without regularlity assumptions.
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