Low Degree Testing over the Reals

We study the problem of testing whether a function f : Rⁿ → R is a polynomial of degree at most d in the distribution-free testing model. Here, the distance between functions is measured with respect to an unknown distribution D over Rⁿ from which we can draw samples. In contrast to previous work, we do not assume that D has finite support. We design a tester that given query access to f, and sample access to D, makes poly(d/ε) many queries to f, accepts with probability 1 if f is a polynomial of degree d, and rejects with probability at least 2/3 if every degree-d polynomial P disagrees with f on a set of mass at least ε with respect to D. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to f, or when f can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.