Proof of Nullstellensatz certificate degree
The research area relevant to your question is propositional proof complexity, more specifically algebraic proof systems. Nullstellensatz is in fact only one of a family of proof systems which includes also the polynomial calculus (PC), polynomial calculus with resolution (PCR), and a proof system which has received some notice from a surprising directions, the sum of squares (SoS) system.

Unfortunately, I wasn't able to find a recent survey of algebraic proof systems. Toni Pitassi has an old survey, but there are many results not mentioned there. There is some material in the more general survey by Paul Beame and Toni Pitassi. Some more recent results are referenced in a lecture by Pavel Pudlák.