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Thanks to missions like Kepler and TESS, we now have access to tens ofthousands of high precision, fast cadence, and long baseline stellarphotometric observations. In principle, these light curves encode a vast amountof information about stellar variability and, in particular, about thedistribution of starspots and other features on their surfaces. Unfortunately,the problem of inferring stellar surface properties from a rotational lightcurve is famously ill-posed, as it often does not admit a unique solution.Inference about the number, size, contrast, and location of spots can thereforedepend very strongly on the assumptions of the model, the regularizationscheme, or the prior. The goal of this paper is twofold: (1) to explore thevarious degeneracies affecting the stellar light curve "inversion" problem andtheir effect on what can and cannot be learned from a stellar surface givenunresolved photometric measurements; and (2) to motivate ensemble analyses ofthe light curves of many stars at once as a powerful data-driven alternative tocommon priors adopted in the literature. We further derive novel results on thedependence of the null space on stellar inclination and limb darkening and showthat single-band photometric measurements cannot uniquely constrain quantitieslike the total spot coverage without the use of strong priors. This is thefirst in a series of papers devoted to the development of novel algorithms andtools for the analysis of stellar light curves and spectral time series, withthe explicit goal of enabling statistically robust inference about theirsurface properties.

Mapping stellar surfaces I: Degeneracies in the rotational light curve problem