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In this note we show that unsatisfiable systems of linear equations with aconstant number of variables per equation over prime finite fields havepolynomial-size constant-degree semi-algebraic proofs of unsatisfiability.These are proofs that manipulate polynomial inequalities over the reals withvariables ranging in $\{0,1\}$. This upper bound is to be put in contrast withthe known fact that, for certain explicit systems of linear equations over thetwo-element field, such refutations require linear degree and exponential sizeif they are restricted to so-called static semi-algebraic proofs, and eventree-like semi-algebraic and sums-of-squares proofs. Our upper bound is a moreor less direct translation of an argument due to Grigoriev, Hirsch andPasechnik (Moscow Mathematical Journal, 2002) who did it for a family of linearsystems of interest in propositional proof complexity. We point out that theirmethod is more general and can be thought of as simulating Gaussianelimination.

A Note on Semi-Algebraic Proofs and Gaussian Elimination over Prime Fields