### Abstract

We consider a number of combinatorial problems in which rational generating functions may be obtained, whose denominators have factors with certain singularities. Specifically, there exist points near which one of the factors is asymptotic to a nondegenerate quadratic. We compute the asymptotics of the coefficients of such a generating function. The computation requires some topological deformations as well as Fourier-Laplace transforms of generalized functions. We apply the results of the theory to specific combinatorial problems, such as Aztec diamond tilings, cube groves, and multi-set permutations.

Original language | English (US) |
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Pages (from-to) | 3127-3206 |

Number of pages | 80 |

Journal | Advances in Mathematics |

Volume | 228 |

Issue number | 6 |

DOIs | |

State | Published - Dec 20 2011 |

### Keywords

- Amoeba
- Aztec diamond
- Cube grove
- Fourier transform
- Fourier-Laplace
- Generalized function
- Hyperbolic polynomial
- Lacuna
- Multivariate generating function
- Quantum random walk
- Random tiling

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Baryshnikov, Y., & Pemantle, R. (2011). Asymptotics of multivariate sequences, part III: Quadratic points.

*Advances in Mathematics*,*228*(6), 3127-3206. https://doi.org/10.1016/j.aim.2011.08.004