I understand how the Kate Polynomial Commitment Scheme Evaluation Proof works however, I don't understand what is the purpose of it?
In general, in a commitment scheme, Peggy commits to message & sends the commitment to Victor. The purpose of a commitment scheme is two fold
Once Peggy commits to a message, then she cannot change it. At a later stage, when the commitment is opened, Victor can check if the commitment matches the message.
When Peggy sends the commitment to Victor, Victor cannot actually see the message till the opening is done.
In the Kate PCS, the message is encoded as a polynomial. After Peggy sends the commitment of the polynomial to Victor. The Opening is going to be when Peggy reveals the polynomial & Victor checks that the commitment matches.
However, the Kate PCS also has an Evaluation Proof - i.e.assuming the polynomial is $f(x)$, then after the commitment is sent to Victor, Victor can ask Peggy to evaluate the polynomial at a value $u$ & prove the evaluation. i.e. if $f(u) = v$, then the evaluation proof proves to Victor that the polynomial originally committed to by Peggy indeed evaluates to $v$ at $u$.
I understood how Kate PCS does this, however I don't understand what's the purpose of this? Of what practical use it for Victor to know the evaluation of the polynomial at one value? I think Kate is used zkSNARKS - but how exactly? In zkSNARKS, the polynomial represents the trace of the transaction. zkSNARKS are non-interactive, so who decides the $u$ at which the polynomial is evaluated & how exactly does it help in verifying the transaction without knowing the transaction. In zkSNARKS, is the evaluation proof of each polynomial which is committed provided at one value $u$ or at multiple values? Though I understand Kate PCS, I am unable to understand at a higher level how it's used in zkSNARKs.