# 함수 다이로그 항등식(functional dilogarithm identity)

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## 개요

• 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화

$L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)$

• 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
• 가령 $$A_n$$ 딘킨 다이어그램으로부터, n 변수로 구성된 $$(n^2+3n)/2$$ 항 관계식을 찾을 수 있음
• $$2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots$$

## 2항 관계식

• $$S=\left\{x,\frac{1}{x}\right\}$$라 두면,

$\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{1}{x}+1}\right)+L\left(\frac{1}{x+1}\right)=L(1)$

## 5항 관계식

• $$S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}$$ 이면,

$\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{x+1}{y}+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)=2L(1)$

## 9항 관계식

• $$S$$를 다음과 같이 두자

$S=\left\{x,y,z,\frac{(x+1) (z+1)}{y},\frac{(x+y+1) (y+z+1)}{x y z},\\ \frac{x z+x+y+z+1}{x y},\frac{x z+x+y+z+1}{y z},\frac{y+1}{x},\frac{y+1}{z}\right\}$

• 다이로그 함수에 대하여 다음이 성립한다

$\sum_{a\in S}L(\frac{1}{1+a})=3L(1)$

## 14항 관계식

$\left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y}, \frac{(w+z+1) (x z+x+y+z+1)}{w y z}\\,\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\\ \frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z}, \frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\}$ $\sum_{a\in S}L(\frac{1}{1+a})=4L(1)$

## 관련논문

• Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777
• Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869.
• Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459.