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We consider the graph $k$-colouring problem encoded as a set of polynomialequations in the standard way over $0/1$-valued variables. We prove that thereare bounded-degree graphs that do not have legal $k$-colourings but for whichthe polynomial calculus proof system defined in [Clegg et al '96, Alekhnovichet al '02] requires linear degree, and hence exponential size, to establishthis fact. This implies a linear degree lower bound for any algorithms based onGr\"{o}bner bases solving graph $k$-colouring using this encoding. The samebound applies also for the algorithm studied in a sequence of papers [De Loeraet al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightlydifferent encoding, thus resolving an open problem mentioned in [De Loera et al'08,'09,'11] and [Li '16]. We obtain our results by combining the polynomialcalculus degree lower bound for functional pigeonhole principle (FPHP) formulasover bounded-degree bipartite graphs in [Mik\v{s}a and Nordstr\"{o}m '15] witha reduction from FPHP to $k$-colouring derivable by polynomial calculus inconstant degree.

Graph Colouring is Hard for Algorithms Based on Hilbert's Nullstellensatz and Gr\"{o}bner Bases