We consider the graph $k$-colouring problem encoded as a set of polynomial equations in the standard way over $0/1$-valued variables. We prove that there are bounded-degree graphs that do not have legal $k$-colourings but for which the polynomial calculus proof system defined in [Clegg et al '96, Alekhnovich et al '02] requires linear degree, and hence exponential size, to establish this fact. This implies a linear degree lower bound for any algorithms based on Gr\"{o}bner bases solving graph $k$-colouring using this encoding. The same bound applies also for the algorithm studied in a sequence of papers [De Loera et al '08,'09,'11,'15] based on Hilbert's Nullstellensatz proofs for a slightly different 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 polynomial calculus degree lower bound for functional pigeonhole principle (FPHP) formulas over bounded-degree bipartite graphs in [Mik\v{s}a and Nordstr\"{o}m '15] with a reduction from FPHP to $k$-colouring derivable by polynomial calculus in constant degree.

翻译：考虑将图$k$-着色问题标准编码为关于$0/1$值变量的多项式方程组。我们证明存在一类有界度图，虽然不存在合法的$k$-着色方案，但[Clegg等人 1996, Alekhnovich等人 2002]定义的微积分证明系统需要线性度数（因而指数级规模）才能建立这一事实。这意味着基于格罗布纳基求解该编码下$k$-着色问题的任何算法都存在线性度数下界。同样的下界也适用于[De Loera等人 2008,2009,2011,2015]系列论文中基于希尔伯特零点定理证明（采用略有不同的编码）的算法，从而解决了[De Loera等人 2008,2009,2011]和[Li 2016]中提出的开放问题。我们的结果通过将[Mikša和Nordström 2015]中有界度二分图上泛函鸽巢原理公式的微积分度数下界与从FPHP到$k$-着色问题的常数度可微积分归约相结合而获得。