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In this paper, we are concerned with the S-polyregularity the regular dotproduct of slice regular functions. We prove that the product of a sliceregular function and right quaternionic polynomial function is a S-polyregularfunction and we determinate its exact order. The general case of the product ofany two slice regular functions is also discussed. In fact, we providesufficient and necessary conditions to the product of slice regular functionsbe a S-polyregular function. The obtained results are then extended to theproduct of S-polyregular functions and remain valid for a special dot product.As consequences we obtain linearization theorems for such S-polyregularproducts with respect to the slice regular functions.

Polyregularity of the dot product of slice regular functions