Projective Quotient Polynomials and Kernel-Weight Layers over Finite Fields Calculations related to Question 2 of Beierle

Let L and M be q-polynomials on Fqn , where q is an arbitrary prime power. We study therelationship between their quotient-image sets and the dimensions of the kernels of L − aXand M − aX. A projective product records the integer multiplicities of the quotient values.Its Moore-determinant form yields principal-minor coefficients with Frobenius recurrences.These relations, together with Newton’s identities in characteristic p, determine the coefficientsfrom the quotient-image set and in turn determine each kernel dimension. The associateddifferential-spectrum calculation for x 7 → xL(x) is included.