2019/09/02 by Malkoun, Joseph
#14N20 #15A15 #51Fxx #51M10 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1909.00571
The Atiyah-Sutcliffe normalized determinant function D is a smooth complex-valued function on Cn(H3), where Cn(H3) denotes the configuration space of n distinct points in hyperbolic 3-space H3. The hyperbolic version of the Atiyah-Sutcliffe conjecture 1 (AS conjecture 1) states that D is nowhere vanishing. AS conjecture 2 (hyperbolic version) is the stronger statement that |D(x)| ≥ 1 for any x ∈ Cn(H3). In this short article, we prove AS conjecture 2 for hyperbolic convex coplanar quadrilaterals, that is for configurations of 4 points in H2 with none of the points in the configuration lying in the convex hull of the other three. We also obtain Y. Zhang and J. Ma's result, namely AS conjecture 1 for non-convex quadrilaterals in H2. Finally, we find an explicit lower bound for |D| depending on n only for the natural ``star-based'' variant of the AS problem, for convex coplanar hyperbolic configurations. The latter result holds for any n ≥ 2. The proofs for n=4 make use of the symbolic library of Python. The proof of the general result follows from a general formula for the determinant. In all these cases, D can be expanded as a linear combination of non-negative rational functions with positive coefficients.