Let a(x) and b(x) be two polynomials. I would like to prove that the GCD of the two polynomials is 1. Do you know if there is a special scheme that proves such statement (i.e., not a NIZK for a general computation such as zk-SNARK)?

Is there perhaps a simpler version that only proves (in zero knowledge) that GCD(a,b)=d for public integers a,b,d?