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We study the computational efficiency of approaches, based on Hilbert'sNullstellensatz, which use systems of linear equations for detectingnon-colorability of graphs having large girth and chromatic number. We showthat for every non-$k$-colorable graph with $n$ vertices and girth $g>4k$, thealgorithm is required to solve systems of size at least $n^{\Omega(g)}$ inorder to detect its non-$k$-colorability.

Graphs with large girth and chromatic number are hard for Nullstellensatz