Counting solutions without zeros or repetitions of a linear congruence and rarefaction in b-multiplicative sequences

Abstract

Consider a strongly $b$-multiplicative sequence and a prime $p$. Studying its $p$-rarefaction consists in characterizing the asymptotic behaviour of the sums of the first terms indexed by the multiples of $p$. The integer values of the "norm" $3$-variate polynomial $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)\!:=\!\prod_{j=1}^{p-1}\left(Y_0{+}\zeta_p^{i_1j}Y_1{+}\zeta_p^{i_2j}Y_2\right),$ where $\zeta_p$ is a primitive $p$-th root of unity, and $i_1,i_2{\in}\{1,2,\dots,p{-}1\},$ determine this asymptotic behaviour. It will be shown that a combinatorial method can be applied to $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2).$ The method enables deducing functional relations between the coefficients as well as various properties of the coefficients of $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$, in particular for $i_1{=}1,i_2{=}2,3.$ This method provides relations between binomial coefficients. It gives new proofs of the two identities $\prod_{j=1}^{p-1}\left(1{-}\zeta_p^j\right){=}p$ and $\prod_{j=1}^{p-1}\left(1{+}\zeta_p^j{-}\zeta_p^{2j}\right){=}L_p$ (the $p$-th Lucas number). The sign and the residue modulo $p$ of the symmetric polynomials of $1{+}\zeta_p{-}\zeta_p^2$ can also be obtained. An algorithm for computation of coefficients of $\mathcal N_{p,i_1,i_2}(Y_0,Y_1,Y_2)$ is developed.