Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests.

In this paper we study a test introduced by Ben-Sasson and Sudan in 2006 that only considers restrictions along axis-parallel hyperplanes. While such a test is necessarily "weaker", it works for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests.

(1) Bivariate low-degree testing with low-agreement.
We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman (1994) in the low-agreement regime, albeit with much larger field size. Namely, for the 2-wise tensor product of the Reed--Solomon code, we prove that for sufficiently large fields, the 2-query variant of the axis-parallel line test (row-vs-column test) works for arbitrarily small agreement. Prior analyses of axis-parallel tests assumed high agreement, and no results for such tests in the low-agreement regime were known.

Our proof technique deviates significantly from that of Polishchuk and Spielman, which relies on algebraic methods such as Bézout's Theorem, and instead leverages a fundamental result in extremal graph theory by Kövári, Sós, and Turán. To our knowledge, this is the first time this result is used in the context of low-degree testing.

(2) Improved robustness for tensor product codes.
Robustness is a strengthening of local testability that underlies many applications. We prove that the axis-parallel hyperplane test for the m-wise tensor product of a linear code with block length n and distance d is Omega(d^m/n^m)-robust. This improves on a theorem of Viderman (2012) by a factor of 1/\poly(m). While the improvement is not large, we believe that our proof is a notable simplification compared to prior work.