Manuel E. Lladser: What Is the Order of $a_{r,r+o(r)}$, When $a_{r,r}$ Decays Polynomially but $a_{r,d\cdot r}$ ($d\ne1$) Decays Exponentially?

Question like the title of our talk are of crucial relevance to fulfill the asymptotic description of certain two-dimensional arrays of numbers. We motivate the question through an example relating cube-roots asymptotics, and also connect it to rediscover the Airy Phenomena. In both instances, asymptotics for the corresponding coefficients relate to asymptotics for a "degenerate" Fourier-Laplace integral. By this we mean that the phase term does not preserve its degree of vanishing (at its stationary points) as $r\to\infty$. The examples we discuss are types of a 2-to-3 change of degree -- that is, the phase term vanishes to degree 2 for finite-r, however, in the limit, it vanishes to degree 3. We illustrate a general technique to handle the leading term of such integrals and discuss the unexpected existence of asymptotic stationary points.

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