Fixed Polarity Pascal Transforms with Symbolic Computer Algebra Applications

Abstract

The fixed polarity forms of the Reed-Muller (RM) transform exist in 2n different polarities. The integer-valued Pascal transform is related to the binary-valued RM transform through the Sierpinski fractal, calculated by performing the modulo-2 operation on Pascal’s triangle, as it appears in the lower triangular portion of the positive-polarity RM transform. We generalize the relationship between the fixed-polarity forms of the RM transform and introduce associated forms of the Pascal transform that are characterized by a polarity value allowing for a family of fixed-polarity Pascal (FPP) transform matrices to be defined. We observe and prove several properties of the FPP transforms and their inverses. An application of FPP transforms in the area of symbolic computer algebra that enables very fast decomposition of real-valued polynomials as weighted sums of different binomials raised to a power as compared to manual symbolic manipulation is described. The decomposition weights can be considered to be the inverse FPP spectrum with respect to a real-valued polynomial since they are computed using one of the linear orthogonal FPP transformation matrices.