## Duke Mathematical Journal

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

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

Li Guo, Sylvie Paycha, and Bin Zhang

#### 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**

Received: 12 March 2015

Revised: 20 April 2016

First available in Project Euclid: 9 November 2016

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-3715303

**Mathematical Reviews number (MathSciNet)**

MR3606725

**Zentralblatt MATH identifier**

1373.52020

**Subjects**

Primary: 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07] 52C07: Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]

Secondary: 65B15: Euler-Maclaurin formula 11M32: Multiple Dirichlet series and zeta functions and multizeta values

**Keywords**

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

#### 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