The $(t,n)$ Shamir’s polynomial based secret sharing scheme is $(+,+)$-homomorphic in which the addition of two polynomials secrets equals the Lagrange’s interpolation of the sum-of-shares for the same subset of shares.
My question is: Does the two polynomials need to have the same degree to satisfy the SSS $(+,+)$-homomorphic property? Specifically, suppose that polynomial $P_1$ defines secret $s_1$ and polynomial $P_2$ defines secret $s_2$. The first polynomial $P_1$ is of $(t-1)$-degree and the second polynomial $P_2$ is $(m-1)$-degree in which $m\ge t$. Suppose I have $m$ shareholders. In this case, can I still say that SSS is $(+,+)$-homomorphic for the same subset of $m$ shares?