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15 February 2017 Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones
Li Guo, Sylvie Paycha, Bin Zhang
Duke Math. J. 166(3): 537-571 (15 February 2017). DOI: 10.1215/00127094-3715303

## Abstract

We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties—reminiscent of the inclusion-exclusion principle for the cardinal on finite sets—of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler–Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler–Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.

## Citation

Li Guo. Sylvie Paycha. Bin Zhang. "Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones." Duke Math. J. 166 (3) 537 - 571, 15 February 2017. https://doi.org/10.1215/00127094-3715303

## Information

Received: 12 March 2015; Revised: 20 April 2016; Published: 15 February 2017
First available in Project Euclid: 9 November 2016

zbMATH: 1373.52020
MathSciNet: MR3606725
Digital Object Identifier: 10.1215/00127094-3715303

Subjects:
Primary: 11H06 , 52B20 , 52C07
Secondary: 11M32 , 65B15

Keywords: algebraic Birkhoff factorization , coalgebras , convex cones , Euler–Maclaurin formula , meromorphic functions , subdivision property