## Duke Mathematical Journal

### Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones

#### 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.

#### Article information

Source
Duke Math. J., Volume 166, Number 3 (2017), 537-571.

Dates
Revised: 20 April 2016
First available in Project Euclid: 9 November 2016

https://projecteuclid.org/euclid.dmj/1478660420

Digital Object Identifier
doi:10.1215/00127094-3715303

Mathematical Reviews number (MathSciNet)
MR3606725

Zentralblatt MATH identifier
1373.52020

#### Citation

Guo, Li; Paycha, Sylvie; Zhang, Bin. Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones. Duke Math. J. 166 (2017), no. 3, 537--571. doi:10.1215/00127094-3715303. https://projecteuclid.org/euclid.dmj/1478660420