# An invariant property of Mahler measure

Matilde Lalín  and  Siva Sankar Nair  Matilde Lalín: Département de
mathématiques et de statistique, Université de Montréal, CP 6128, succ.
Centre-ville, Montreal, QC H3C 3J7, Canada<EMAIL_ADDRESS>Siva
Sankar Nair: Département de mathématiques et de statistique, Université de
Montréal, CP 6128, succ. Centre-ville, Montreal, QC H3C 3J7, Canada
<EMAIL_ADDRESS>
###### Abstract.

We exhibit a change of variables that maintains the Mahler measure of a given
polynomial. This method leads to the construction of highly non-trivial
polynomials with given Mahler measure and settles some conjectural numerical
formulas due to Boyd and Brunault.

###### Key words and phrases:

Mahler measure; zeta values

###### 2010 Mathematics Subject Classification:

Primary 11R06; Secondary 11M06, 11R42

##  1\. Introduction

Given a non-zero rational function $P\in\mathbb{C}(x_{1},\dots,x_{n})$, its
(logarithmic) Mahler measure is given by

$\mathrm{m}(P):=\frac{1}{(2\pi
i)^{n}}\int_{\mathbb{T}^{n}}\log|P(x_{1},\dots,x_{n})|\frac{\mathrm{d}x_{1}}{x_{1}}\cdots\frac{\mathrm{d}x_{n}}{x_{n}},$

where the integration is taken over the unit torus
$\mathbb{T}^{n}=\\{(x_{1},\dots,x_{n})\in\mathbb{C}^{n}:|x_{1}|=\cdots=|x_{n}|=1\\}$
with respect to the Haar measure.

This quantity first appeared (for one variable-polynomials) in considerations
by Lehmer [Leh33] in his work related to the generalization of Mersenne
numbers, and was later extended to polynomials with several variables by
Mahler [Mah62] in applications to polynomial heights. About 20 years later
Smyth [Smy81, Boy81] proved the formulas

$\mathrm{m}(x+y+1)=\frac{3\sqrt{3}}{4\pi}L(\chi_{-3},2),$ (1)
$\mathrm{m}(x+y+z+1)=\frac{7}{2\pi^{2}}\zeta(3),$

where $L(\chi_{-3},s)$ is the Dirichlet $L$-function in the character of
conductor 3 and $\zeta(s)$ is the Riemann zeta function.

These formulas opened the door to a wave of research in Mahler measure and
special values of functions with arithmetical significance. Consider, for
example, the following formula conjectured by Deninger [Den97] and Boyd
[Boy98] and proven by Rogers and Zudilin [RZ14]:

$\mathrm{m}\left(x+\frac{1}{x}+y+\frac{1}{y}+1\right)=L^{\prime}(E_{15a8},0),$

where $L(E_{15a8},s)$ denotes the $L$-function associated to the elliptic
curve $15a8$.

The appearance of values of $L$-functions in certain Mahler measure formulas
was explained by Deninger in terms of Beilinson’s conjectures via
relationships with regulators. (See also the works of Boyd [Boy98] and
Rodriguez-Villegas [RV99] for additional insights, and the book of Brunault
and Zudilin [BZ20] for more details.) Likewise, the cases involving the
Riemann zeta function and Dirichlet $L$-functions have been linked to
particular applications of the Borel regulator and evaluations of
polylogarithms in algebraic numbers [BRV02, BRVD03, Lal07, Lal08].

A particularly interesting formula was proven by Condon [Con04]:

(2) $\mathrm{m}(x+1+(x-1)(y+z))=\frac{28}{5\pi^{2}}\zeta(3).$

Although this identity seems similar to (1), it is indeed much harder to
prove. While the right-hand side of (1) is the result of evaluating the
trilogarithm in $1$ and $-1$, the right-hand side of (2) is the result of
evaluating the trilogarithm in combinations of $\varphi=\frac{1+\sqrt{5}}{2}$.
This complicates the computation considerably, independently of whether one
uses elementary methods as in [Con04] or the regulator as in [Lal07].

In [Boy06], Boyd proposed the numerical study of polynomials of the form
$a(x)+b(x)y+c(x)z$, where $a(x),b(x),c(x)$ are products of cyclotomic
polynomials. The focus on this particular class of polynomials was motivated
by the Cassaigne–Maillot formula for the Mahler measure of $a+by+cz$ [Mai00],
which has an expression that is specially well-suited for numerical
integration. The exploration of such polynomials led to the discovery of
several interesting numerical formulas involving $L^{\prime}(E,-1)$ for
various elliptic curves [Lal15]. For example, Boyd discovered

(3)
$\mathrm{m}(1+(x-1)y+(x+1)z)\stackrel{{\scriptstyle?}}{{=}}\frac{5}{4}L^{\prime}(E_{21a1},-1),$

where the question mark denotes a numerical identity that has been verified to
at least 20 decimal places, and also

(4)
$\mathrm{m}(x^{2}+x+1+(x^{2}-1)(y+z))\stackrel{{\scriptstyle?}}{{=}}\frac{28}{5\pi^{2}}\zeta(3),$

which involves, again, the term $\frac{28}{5\pi^{2}}\zeta(3)$.

More recently Brunault further pursued these computations (with higher degree
cyclotomic polynomials) and discovered various numerical formulas yielding
$\frac{28}{5\pi^{2}}\zeta(3)$ such as

(5) $\displaystyle\mathrm{m}(x^{4}-x^{3}+x^{2}-x+1+(x^{4}-x^{3}+x-1)(y+z))$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{28}{5\pi^{2}}\zeta(3),$ (6)
$\displaystyle\mathrm{m}(x^{5}+x^{4}+x+1+(x^{5}-1)(y+z))$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{28}{5\pi^{2}}\zeta(3),$

and several others.

We started this project by investigating whether the above polynomials having
Mahler measure $\frac{28}{5\pi^{2}}\zeta(3)$ had anything in common with each
other. We discovered that they all could be obtained from Condon’s polynomial
in (2) by certain changes of variables. In fact, such changes of variables
provide a method for generating arbitrarily many rational functions that have
the same Mahler measure, giving rise to highly non-trivial identities.
Moreover, we can also obtain families of conjectures by applying the change of
variables on polynomials such as the one in (3).

Before stating our main result, we establish some notation. For a polynomial
$g(x)\in\mathbb{C}[x]$, such that $g(x)=\sum_{j=0}^{d}g_{j}x^{j}$ with
$g_{d}\not=0$, denote by

$\overline{g}(x)=\sum_{j=0}^{d}\overline{g}_{j}x^{j},$

the polynomial resulting from conjugating the coefficients of $g(x)$.

We prove the following result.

######  Theorem 1.

Let $P(x,y_{1},\dots,y_{n})$ be a polynomial over $\mathbb{C}$ in the
variables $x,y_{1},\dots,y_{n}$. Let $g(x)\in\mathbb{C}[x]$ be a polynomial in
the variable $x$ with all roots outside the unit disc, and for an integer $k$
greater than the degree of $g(x)$ and a complex number $\lambda$ of absolute
value one, let $f(x)=\lambda x^{k}\overline{g}(x^{-1})$. We denote by
$\widetilde{P}$ the rational function obtained by replacing $x$ by $f(x)/g(x)$
in $P$. Then

$\mathrm{m}(P)=\mathrm{m}(\widetilde{P}).$

Using this theorem, equations (4), (5), and (6) follow by taking
$P(x,y,z)=x+1+(x-1)(y+z)$, which is the polynomial from Condon’s result in
(2), with $g(x)=x+2,x^{2}-2x+2,$ and $x^{4}+x+2$ along with $k=2,4,$ and $5$
respectively, and $\lambda=1$. We have a term of $\mathrm{m}(g)=\log 2$ in all
these cases, which cancels with the contribution from a factor of $2$
occurring in $\widetilde{P}$, leading us to the desired identities.

As another example we can prove the following formulas

(7) $\displaystyle\mathrm{m}((x^{2}+x+1)(1+w)(1+u)z+(x^{2}-1)(1-w)(1+y))=$
$\displaystyle\frac{93}{\pi^{4}}\zeta(5),$ (8)
$\displaystyle\mathrm{m}((x^{4}-x^{3}+x^{2}-x+1)(1+w)(1+u)z+(x^{4}-x^{3}+x-1)(1-w)(1+y))=$
$\displaystyle\frac{93}{\pi^{4}}\zeta(5),$ (9)
$\displaystyle\mathrm{m}((x^{5}+x^{4}+x+1)(1+w)(1+u)z+(x^{5}-1)(1-w)(1+y))=$
$\displaystyle\frac{93}{\pi^{4}}\zeta(5),$

by applying Theorem 1 to a result proven in [Lal06].

In addition, by applying this method to (3), we obtain a family of conjectures
involving $L^{\prime}(E_{21a1},-1)$, including

(10)
$\displaystyle\mathrm{m}\left(\frac{x+2}{2}+(x^{2}+x+1)y+(x^{2}-1)z\right)$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{5}{4}L^{\prime}(E_{21a1},-1),$
(11)
$\displaystyle\mathrm{m}\left(\frac{x^{2}-2x+2}{2}+(x^{4}-x^{3}+x^{2}-x+1)y+(x^{4}-x^{3}+x-1)z\right)$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{5}{4}L^{\prime}(E_{21a1},-1),$
(12)
$\displaystyle\mathrm{m}\left(\frac{x^{4}+x+2}{2}+(x^{5}+x^{4}+x+1)y+(x^{5}-1)z\right)$
$\displaystyle\stackrel{{\scriptstyle?}}{{=}}\frac{5}{4}L^{\prime}(E_{21a1},-1).$

This article is organized as follows. In Section 2 we consider some results
needed in preparation for the proof of Theorem 1, which is included in Section
3. In Section 4 we discuss some applications of Theorem 1, mainly to families
of rational functions with an arbitrary number of variables. Finally, Section
5 sketches the relation between Mahler measures and regulators, with the goal
of giving more context to some of the new formulas proven in this article that
involve polynomials defining curves of high genus.

## Acknowledgements

The authors would like to thank David Boyd, François Brunault, Andrew
Granville, and Wadim Zudilin for helpful discussions, and also gratefully
acknowledge the support of Sujatha Ramdorai. The authors are thankful to the
referee for their helpful remarks. This work is supported by the Natural
Sciences and Engineering Research Council of Canada, Discovery Grant
355412-2022, the Fonds de recherche du Québec - Nature et technologies, Projet
de recherche en équipe 300951, the Institut des sciences mathématiques and the
Centre de recherches mathématiques.

##  2\. Some intermediate results

Theorem 1 depends on the nature of the change of variables $x\rightarrow
f(x)/g(x)$ and how this affects the Mahler measure. In this section, we
address this question and prove some intermediate results regarding the Mahler
measure of certain univariate polynomials. These results will play an
important role in the proof of the main result later on.

We first present the following lemma concerning the roots of a polynomial
obtained from the polynomials $f(x)$ and $g(x)$ that appear in the change of
variables.

######  Lemma 2.

[Lax44, Lemma 1], [Che95, Theorem 1] Let $g(x)\in\mathbb{C}[x]$ be such that
all its roots have absolute value greater than or equal to one, let $k$ be an
integer such that $k\geq\mathrm{deg}\;(g)$ and let $f(x)=\lambda
x^{k}\overline{g}(x^{-1})$, where $\lambda$ is a complex number with absolute
value one. For any complex number $\beta$, consider

${\Gamma_{\beta}}(x)=f(x)+\beta g(x)\in\mathbb{C}[x].$

Then we have the following:

  1. (a)

If $|\beta|<1$, then all roots of ${\Gamma_{\beta}}(x)$ have absolute value
less than one.

  2. (b)

If $|\beta|>1$, then all roots of ${\Gamma_{\beta}}(x)$ have absolute value
greater than one.

  3. (c)

If $|\beta|=1$, then all roots of ${\Gamma_{\beta}}(x)$ have absolute value
one.

######  Remark.

This result has a rich history, as it was reproven several times. As stated,
the original result is due to Lax [Lax44] (for the case $k=\deg(g)$) and Chen
[Che95] (for the general case). Both consider the case when $|\beta|=1$ in
their statements, but their proofs include the three cases stated in Lemma 2.
This result was later rediscovered by Lalín and Smyth [LS13, Theorem 1]
(stated for the case $k>\deg(g)$). A converse was stated and proven by Cohn
[Coh22]. Further results investigate the interlacing of the roots of
$\Gamma_{\beta}$ as $\beta$ varies in the unit circle, see [LS15] and the
references therein for more information.

The above result is true for any $k$ greater than or equal to $d$, the degree
of $g(x)$. However, for the rest of our results to hold true, we would require
$k$ to be strictly greater than $d$.

The above lemma gives us the following corollary using Jensen’s formula.

######  Corollary 3.

Let $f$ and $g$ be as before with $k$ strictly greater than the degree of
$g(x)$. Then for any $\beta\in\mathbb{C}$, we have

$\mathrm{m}(\Gamma_{\beta}(x))=\log|g_{0}|+\log^{+}|\beta|,$

where $\log^{+}|x|:=\log\max\\{1,|x|\\}$.

###### Proof.

Note that $\Gamma_{\beta}(x)=\lambda x^{k}\overline{g}(x^{-1})+\beta g(x)$,
and since $k>\mathrm{deg}\;g$, the leading coefficient here is
$\lambda\overline{g_{0}}$ and the constant term is $\beta g_{0}$. By Lemma 2,
we know that if $\alpha_{j}$ are the roots of $\Gamma_{\beta}$, then either
$|\alpha_{j}|\geq 1$ for all $j$, or $|\alpha_{j}|\leq 1$ for all $j$. In both
cases, this means that by Jensen’s formula,

$\displaystyle\mathrm{m}(\Gamma_{\beta})-\log|\lambda\overline{g_{0}}|=\sum_{j}\log^{+}|\alpha_{j}|$
$\displaystyle=\log^{+}\Big{|}\prod_{j}\alpha_{j}\Big{|}$
$\displaystyle=\log^{+}\left|\frac{\beta
g_{0}}{\lambda\overline{g_{0}}}\right|=\log^{+}|\beta|,$

and since $\log|\lambda\overline{g_{0}}|=\log|g_{0}|$, we get the desired
result. ∎

The next statement is essentially a restatement of Corollary 3 in slightly
more general terms. It will be used to prove Theorem 1 in the next section.

######  Proposition 4.

Let $f$ and $g$ be as before with $k$ strictly greater than the degree of
$g(x)$. Then for any $\alpha,\beta\in\mathbb{C}$, not both zero, we have

$\mathrm{m}(\alpha f+\beta g)=\mathrm{m}(g)+\log\max\\{|\alpha|,|\beta|\\}.$

######  Remark.

Note that since $g$ has all its roots outside the unit circle, we have
$\mathrm{m}(g)=\log|g_{0}|$. Similarly, since $f$ has all its roots inside the
unit circle, its Mahler measure is given by its leading coefficient, which is
$\lambda\overline{g_{0}}$. Thus, $\mathrm{m}(f)=\mathrm{m}(g)=\log|g_{0}|.$

###### Proof.

Consider

$\Gamma(x)=\alpha f(x)+\beta g(x)=\lambda\alpha
x^{k}\overline{g}(x^{-1})+\beta g(x).$

Then $\deg(\Gamma)=k$ as before. We first consider the case $\alpha\neq 0$.
Without loss of generality, we can then assume that $\alpha=1$. Using
Corollary 3 with $\Gamma_{\beta}(x)=\Gamma(x)$, we have

$\displaystyle\mathrm{m}(\Gamma(x))$
$\displaystyle=\log|g_{0}|+\log^{+}|\beta|$
$\displaystyle=\mathrm{m}(g)+\log\max\\{1,|\beta|\\}$
$\displaystyle=\mathrm{m}(g)+\log\max\\{|\alpha|,|\beta|\\},$

as needed. Coming to the second case where $\alpha=0$, then

$\mathrm{m}(\alpha f+\beta g)=\mathrm{m}(\beta g)=\mathrm{m}(g)+\log|\beta|,$

and since $\log\max\\{|\alpha|,|\beta|\\}=\log|\beta|$, we get the desired
result. ∎

Before moving on to the proof of the main theorem, we will also briefly
discuss the existence of the Mahler measure integral for any polynomial in
general. Although a fundamental and well-known fact, we prove the result here
for completion, due to its central role in the proof of Theorem 1.

Let $F(y_{1},y_{2},\dots,y_{n+1})$ be a polynomial in
$\mathbb{C}[y_{1},\dots,y_{n+1}]$ in $(n+1)$ variables. We may write $F$ as a
polynomial in $y_{n+1}$ with coefficients being $n$-variable polynomials
$c_{i}\in\mathbb{C}[y_{1},\dots,y_{n}]$, as

$F=c_{0}+c_{1}y_{n+1}+\cdots+c_{k}y_{n+1}^{k},$

for some non-negative integer $k$. We may also factorize $F$ as

$F=c_{k}\prod_{j=1}^{k}\Big{(}y_{n+1}-\Delta_{j}(y_{1},y_{2},\dots,y_{n})\Big{)},$

where $\Delta_{j}(y_{1},y_{2},\dots,y_{n})$ are algebraic functions in
$y_{1},\dots,y_{n}$ with appropriately chosen branch cuts. Now consider the
following lemma:

######  Lemma 5.

[EW99, Lemma 3.7] Let $F$ be a polynomial as above. Then the Mahler measure
$\mathrm{m}(F)$ of $F$ exists and we can write

$\mathrm{m}(F)=\mathrm{m}(c_{k})+\frac{1}{(2\pi
i)^{n}}\sum_{j=1}^{k}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}},$

where all integrals involved converge.

###### Proof.

We proceed by induction on the number of variables in the polynomial $F$. In
the base case where $n=1$, the $\Delta_{j}$’s are simply the roots of $F$, and
the result follows by Jensen’s formula. Now assume the result is true for $n$
variables. Using the factorization of $F$, we can write

$\displaystyle\mathrm{m}(F)$ $\displaystyle=\frac{1}{(2\pi
i)^{n+1}}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\int_{|y_{n+1}|=1}\log|F|\;\frac{\mathrm{d}y_{n+1}}{y_{n+1}}\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}$
$\displaystyle=\mathrm{m}(c_{k})+\frac{1}{(2\pi
i)^{n+1}}\sum_{j=1}^{k}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\underbrace{\Big{(}\int_{|y_{n+1}|=1}\log\Big{|}y_{n+1}-\Delta_{j}\Big{|}\;\frac{\mathrm{d}y_{n+1}}{y_{n+1}}\Big{)}}_{I}\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}.$

Recall that if $\beta\in\mathbb{C}$, then Jensen’s formula implies that

$\mathrm{m}(x+\beta)=\frac{1}{2\pi
i}\int_{|x|=1}\log|x+\beta|=\log^{+}|\beta|.$

Thus, for constant $y_{1},y_{2},\dots,y_{n}$, the inner integral above is
given by $I=2\pi i\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|$, and we have

(13) $\displaystyle\mathrm{m}(F)=\mathrm{m}(c_{k})+\frac{1}{(2\pi
i)^{n}}\sum_{j=1}^{k}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}.$

The polynomial $c_{k}$ has $n$ variables and by the induction hypothesis, it
follows that $\mathrm{m}(c_{k})$ exists. It remains to show that the integral
in (13) also exists. First, we note that the torus $\mathbb{T}^{n+1}$ defined
by $\\{(y_{1},y_{2},\dots,y_{n+1})\in\mathbb{C}^{n+1}\;:\;|y_{j}|=1\\}$ is
compact, and hence the polynomial $F$ must be bounded above on this region,
and therefore $\mathrm{m}(F)$ is bounded. Since $\mathrm{m}(c_{k})$ is finite,
this means the integral in (13) is also bounded from above. Secondly, if we
write $y_{j}=e^{i\theta_{j}}$ for a real variable $\theta_{j}$, then

$\frac{\mathrm{d}y_{j}}{y_{j}}=i\;\mathrm{d}\theta_{j},$

for each $j$. Taking into account the factor of $(i)^{n}$ in the denominator
outside the summation, we observe that the integrand is
$\log^{+}|\Delta_{j}|$, which is totally real and non-negative. In addition,
although the algebraic functions $\Delta_{j}$ may not be continuous
individually, the multiset of values $\\{\Delta_{j}(y_{1},\dots,y_{n})\\}$
depends continuously on $(y_{1},\dots,y_{n})$, away from the poles of the
$\Delta_{j}$. Thus, the integrands are non-negative and continuous, and along
with the fact that the integrals are bounded above, this shows that they all
converge. This completes the induction and we conclude that $\mathrm{m}(F)$
exists and that the integral in (13) converges. ∎

##  3\. Proof of Theorem 1

We are now ready to prove our main result. Recall that we replace the variable
$x$ by $f(x)/g(x)$ in the polynomial $P$ to obtain the rational function
$\widetilde{P}$. We will show that the Mahler measure computation for both $P$
and $\widetilde{P}$ is given by the same integral.

###### Proof of Theorem 1.

We first write $P$ as a polynomial in $x$ with coefficients in
$\mathbb{C}[y_{1},\dots,y_{n}]$ as

$P=P_{0}+P_{1}x+\cdots+P_{\ell}x^{\ell},$

where $P_{j}\in\mathbb{C}[y_{1},\dots,y_{n}]$ for each $1\leq j\leq\ell$. We
can now factorize this as

$P=P_{\ell}\prod_{j=1}^{\ell}\Big{(}x-\Delta_{j}(y_{1},\dots,y_{n})\Big{)},$

where the $\Delta_{j}$’s are algebraic functions in $y_{1},\dots,y_{n}$ with
appropriately chosen branch cuts. Then, using Lemma 5, the Mahler measure of
$P$ is given by

(14) $\displaystyle\mathrm{m}(P)=\mathrm{m}(P_{\ell})+\frac{1}{(2\pi
i)^{n}}\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}.$

We now move on to the evaluation of $\mathrm{m}(\widetilde{P})$, the rational
function obtained from $P$ by replacing $x$ by $f(x)/g(x)$, which gives

$\displaystyle\widetilde{P}$
$\displaystyle=P_{\ell}\prod_{j=1}^{\ell}\left(\frac{f(x)}{g(x)}-\Delta_{j}(y_{1},\dots,y_{n})\right)$
$\displaystyle=(g(x))^{-\ell}P_{\ell}\prod_{j=1}^{\ell}\Big{(}{f(x)}-g(x)\Delta_{j}(y_{1},\dots,y_{n})\Big{)}.$

Thus, the Mahler measure of $\widetilde{P}$ is given by

$\displaystyle\mathrm{m}(\widetilde{P})$ $\displaystyle=\frac{1}{(2\pi
i)^{n+1}}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\int_{|x|=1}\log|\widetilde{P}|\;\frac{\mathrm{d}x}{x}\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}$
$\displaystyle=\mathrm{m}(P_{\ell})-\ell\cdot\mathrm{m}(g)+\frac{1}{(2\pi
i)^{n+1}}\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\underbrace{\Big{(}\int_{|x|=1}\log\Big{|}{f(x)}-g(x)\Delta_{j}\Big{|}\;\frac{\mathrm{d}x}{x}\Big{)}}_{J}\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}.$

Keeping the $y_{1},y_{2},\dots,y_{n}$ constant and using Proposition 4, we can
write the inner integral $J$ as

$J=\int_{|x|=1}\log\Big{|}{f(x)}-g(x)\Delta_{j}\Big{|}\;\frac{\mathrm{d}x}{x}=2\pi
i\,\mathrm{m}(f-g\Delta_{j})=2\pi
i\Big{(}\mathrm{m}(g)+\log^{+}{\Delta_{j}}\Big{)},$

for each $1\leq j\leq\ell$. Noting that

$\displaystyle\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\Big{(}2\pi
i\,\mathrm{m}(g)\Big{)}\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}$
$\displaystyle=2\pi
i\,\mathrm{m}(g)\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}1\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}$
$\displaystyle=2\pi i\,\mathrm{m}(g)\sum_{j=1}^{\ell}(2\pi
i)^{n}=\ell\cdot\mathrm{m}(g)\cdot(2\pi i)^{n+1},$

we can rewrite the Mahler measure of $\widetilde{P}$ as

$\displaystyle\mathrm{m}(\widetilde{P})$
$\displaystyle=\mathrm{m}(P_{\ell})-\ell\cdot\mathrm{m}(g)+\ell\cdot\mathrm{m}(g)+\frac{1}{(2\pi
i)^{n}}\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}$
(15) $\displaystyle=\mathrm{m}(P_{\ell})+\frac{1}{(2\pi
i)^{n}}\sum_{j=1}^{\ell}\int_{|y_{1}|=1}\cdots\int_{|y_{n}|=1}\log^{+}|\Delta_{j}(y_{1},\dots,y_{n})|\;\frac{\mathrm{d}y_{n}}{y_{n}}\cdots\frac{\mathrm{d}y_{1}}{y_{1}}.$

We observe that equations (14) and (3) evaluate to the same expression and
conclude that

$\mathrm{m}(P)=\mathrm{m}(\widetilde{P}),$

which completes the proof. ∎

##  4\. Applications of Theorem 1

As remarked in the introduction, by taking $P(x,y,z)=x+1+(x-1)(y+z)$, and by
considering its Mahler measure given in (2), we can deduce equations (4), (5),
and (6) by setting $g(x)=x+2,x^{2}-2x+2,$ and $x^{4}+x+2$ with $k=2,4,$ and
$5$ respectively, with $\lambda=1$, and after subtracting $\log 2$ from both
sides of the evaluated identity in Theorem 1.

Theorem 1 can be applied to very general polynomials $P$ and even to rational
functions. Indeed, to work with rational functions, we can apply Theorem 1 to
the numerator and the denominator individually, and then put them back
together to recover the rational function with the change of variables.

Here we examine an interesting application involving rational functions with
arbitrarily many variables whose Mahler measure can be computed explicitly.
Considering the fact that calculating the exact Mahler measure of
multivariable polynomials is generally hard, let alone those with arbitrarily
many variables, this is an intriguing and rare result.

Let

$\displaystyle R_{m}(x_{1},\dots,x_{m},z):=$ $\displaystyle
z+\left(\frac{1-x_{1}}{1+x_{1}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right),$
$\displaystyle S_{m}(x_{1},\dots,x_{m},x,y,z):=$
$\displaystyle(1+x)z+\left(\frac{1-x_{1}}{1+x_{1}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)(1+y),$
$\displaystyle T_{m}(x_{1},\dots,x_{m},x,y):=$ $\displaystyle
1+\left(\frac{1-x_{1}}{1+x_{1}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)x+\left(1-\left(\frac{1-x_{1}}{1+x_{1}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)\right)y.$

For $a_{1},\dots a_{m}\in\mathbb{C}$, define

$s_{\ell}(a_{1},\dots,a_{m})=\left\\{\begin{array}[]{ll}1&\mbox{if }\ell=0,\\\
\sum_{i_{1}<\cdots<i_{\ell}}a_{i_{1}}\cdots a_{i_{\ell}}&\mbox{if }0<\ell\leq
m,\\\ 0&\mbox{if }m<\ell.\end{array}\right.$

Recall that the Bernoulli numbers are given by

$\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}\frac{B_{n}x^{n}}{n!}.$

The Mahler measures of the polynomials $R_{m},S_{m},T_{m}$ can be computed by
the following formulas.

######  Theorem 6.

([Lal06, LL16]) We have the following identities. For $n\geq 1$,

$\mathrm{m}\left(R_{2n}\right)=\sum_{h=1}^{n}\frac{s_{n-h}(2^{2},4^{2},\dots,(2n-2)^{2})}{(2n-1)!}\left(\frac{2}{\pi}\right)^{2h}\mathcal{A}(h),$

where

$\mathcal{A}(h):=(2h)!\left(1-\frac{1}{2^{2h+1}}\right)\zeta(2h+1).$

For $n\geq 0$,

$\mathrm{m}\left(R_{2n+1}\right)=\sum_{h=0}^{n}\frac{s_{n-h}(1^{2},3^{2},\dots,(2n-1)^{2})}{(2n)!}\left(\frac{2}{\pi}\right)^{2h+1}\mathcal{B}(h),$

where

$\mathcal{B}(h):=(2h+1)!L(\chi_{-4},2h+2).$

For $n\geq 1$,

$\mathrm{m}\left(S_{2n}\right)=\sum_{h=1}^{n}\frac{s_{n-h}(2^{2},4^{2},\dots,(2n-2)^{2})}{(2n-1)!}\left(\frac{2}{\pi}\right)^{2h+2}\mathcal{C}(h),$

where

$\mathcal{C}(h):=\sum_{\ell=1}^{h}\binom{2h}{2\ell}\frac{(-1)^{h-\ell}}{4h}B_{2(h-\ell)}\pi^{2h-2\ell}(2\ell+2)!\left(1-\frac{1}{2^{2\ell+3}}\right)\zeta(2\ell+3).$

For $n\geq 0$,

$\mathrm{m}\left(S_{2n+1}\right)=\sum_{h=0}^{n}\frac{s_{n-h}(1^{2},3^{2}\dots,(2n-1)^{2})}{(2n)!}\left(\frac{2}{\pi}\right)^{2h+3}\mathcal{D}(h),$

where

$\mathcal{D}(h):=\sum_{\ell=0}^{h}\binom{2h+1}{2\ell+1}\frac{(-1)^{h-\ell}}{2(2h+1)}B_{2(h-\ell)}\pi^{2h-2\ell}(2\ell+3)!L(\chi_{-4},2\ell+4).$

For $n\geq 1$,

$\mathrm{m}\left(T_{2n}\right)=\frac{\log
2}{2}+\sum_{h=1}^{n}\frac{s_{n-h}(2^{2},4^{2},\dots,(2n-2)^{2})}{(2n-1)!}\left(\frac{2}{\pi}\right)^{2h}\mathcal{E}(h),$

where

$\displaystyle\mathcal{E}(h):=$
$\displaystyle\frac{(2h)!}{2}\left(1-\frac{1}{2^{2h+1}}\right)\zeta(2h+1)+\sum_{\ell=1}^{h}(2^{2(h-\ell)-1}-1)\binom{2h}{2\ell}\frac{(-1)^{h-\ell+1}}{2h}$
$\displaystyle\times
B_{2(h-\ell)}\pi^{2h-2\ell}(2\ell)!\left(1-\frac{1}{2^{2\ell+1}}\right)\zeta(2\ell+1).$

For $n\geq 0$,

$\mathrm{m}\left(T_{2n+1}\right)=\frac{\log
2}{2}+\sum_{h=1}^{n}\frac{s_{n-h}(2^{2},4^{2},\dots,(2n-2)^{2})}{(2n+1)!}\left(\frac{2}{\pi}\right)^{2h+2}\mathcal{F}(h),$

where

$\displaystyle\mathcal{F}(h):=$
$\displaystyle\frac{(2h+2)!}{2}\left(1-\frac{1}{2^{2h+3}}\right)\zeta(2h+3)+\frac{\pi^{2}n^{2}}{2}(2h)!\left(1-\frac{1}{2^{2h+1}}\right)\zeta(2h+1)$
$\displaystyle+n(2n+1)\sum_{\ell=1}^{h}(2^{2(h-\ell)-1}-1)\binom{2h}{2\ell}\frac{(-1)^{h-\ell+1}}{4h}B_{2(h-\ell)}\pi^{2h+2-2\ell}(2\ell)!\left(1-\frac{1}{2^{2\ell+1}}\right)\zeta(2\ell+1).$

We can apply Theorem 1 to each variable $x_{\ell}$, replacing it with
$f(x_{\ell})/g(x_{\ell})$ for any compatible $f(x_{\ell})$, $g(x_{\ell})$.
This yields infinitely many more formulas satisfying the results of Theorem 6.
For example, $g(x_{1})=x_{1}+2$, $k=2$ and $\lambda=1$ gives, for $m\geq 1$

$\displaystyle\mathrm{m}\left(z+\left(\frac{x_{1}^{2}-1}{x_{1}^{2}+x_{1}+1}\right)\left(\frac{1-x_{2}}{1+x_{2}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)\right)=$
$\displaystyle~{}\mathrm{m}\left(R_{m}\right),$
$\displaystyle\mathrm{m}\left((1+x)z+\left(\frac{x_{1}^{2}-1}{x_{1}^{2}+x_{1}+1}\right)\left(\frac{1-x_{2}}{1+x_{2}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)(1+y)\right)=$
$\displaystyle~{}\mathrm{m}\left(S_{m}\right),$

and

$\displaystyle\mathrm{m}\left(1+\left(\frac{x_{1}^{2}-1}{x_{1}^{2}+x_{1}+1}\right)\right.$
$\displaystyle\left.\left(\frac{1-x_{2}}{1+x_{2}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)x\right.$
$\displaystyle\left.+\left(1-\left(\frac{x_{1}^{2}-1}{x_{1}^{2}+x_{1}+1}\right)\left(\frac{1-x_{2}}{1+x_{2}}\right)\cdots\left(\frac{1-x_{m}}{1+x_{m}}\right)\right)y\right)=\mathrm{m}(T_{m}).$

In particular, equation (7) follows from considering the identity with
$S_{2}$. We remark that in this case the formula for the Mahler measure of the
rational function $\widetilde{S_{2}}$ can be directly written as the Mahler
measure of the polynomial in (7) by multiplying by the denominator, which has
Mahler measure zero since it is the product of cyclotomic polynomials.
Similarly, equations (8) and (9) are obtained by using $g(x)=x^{2}-2x+2,$ and
$x^{4}+x+2$ with $k=4,$ and $5$ and $\lambda=1$ with $S_{2}$.

This application is yet another example emphasising the versatility of Theorem
1 towards expanding existing results to generate new identities, that would
otherwise be very difficult to establish.

##  5\. A discussion on the genus of the involved varieties

In this section we briefly discuss the relation of Mahler measure with the
regulator in order to give additional context to the formulas that we have
proven. More specifically, we explain how some of the new polynomials
correspond to curves of higher genus, which motivates our interest in such
formulas.

We start by following Deninger [Den97] as well as the treatment by Brunault
and Zudilin [BZ20, Section 8.3]. Let $F\in\mathbb{C}[y_{1},\dots,y_{n+1}]$ be
an irreducible polynomial in $(n+1)$ variables. As in Section 2, we write

$F=c_{0}+c_{1}y_{n+1}+\cdots+c_{k}y_{n+1}^{k},$

where $c_{i}\in\mathbb{C}[y_{1},\dots,y_{n}]$. Applying Jensen’s formula as in
the proof of Lemma 5, we can write

$\mathrm{m}(F)=\mathrm{m}(c_{k})+\frac{1}{(2\pi
i)^{n}}\int_{D}\log|y_{n+1}|\frac{\mathrm{d}y_{1}}{y_{1}}\wedge\cdots\wedge\frac{\mathrm{d}y_{n}}{y_{n}},$

where

$D=\\{(y_{1},\dots,y_{n})\,:\,|y_{1}|=\cdots=|y_{n}|=1,|y_{n+1}|>1,F(y_{1},\dots,y_{n+1})=0\\}$

is known as the Deninger cycle attached to $F$.

The differential form
$\log|y_{n+1}|\frac{\mathrm{d}y_{1}}{y_{1}}\wedge\cdots\wedge\frac{\mathrm{d}y_{n}}{y_{n}},$
defined over the smooth part $Z_{F}^{\mathrm{reg}}$ of the zero locus of $F$
in $(\mathbb{C}^{\times})^{n+1}$, is not necessarily closed, but there is a
closed form $\eta(y_{1},\dots,y_{n+1})$, defined on the de Rham cohomology of
$Z_{F}^{\mathrm{reg}}$, that satisfies

$\eta(y_{1},\dots,y_{n+1})|_{D}=(-1)^{n}\log|y_{n+1}|\frac{\mathrm{d}y_{1}}{y_{1}}\wedge\cdots\wedge\frac{\mathrm{d}y_{n}}{y_{n}}.$

We have the following theorem.

######  Theorem 7.

[Den97, Proposition 3.3], [BZ20, Theorem 8.11] Let
$F\in\mathbb{C}[y_{1},\dots,y_{n+1}]$ be an irreducible polynomial such that
$\overline{D}$ is a topological $n$-chain contained in $Z_{F}^{\mathrm{reg}}$.
Then

$\mathrm{m}(F)=\mathrm{m}(c_{k})+\frac{(-1)^{n}}{(2\pi
i)^{n}}\int_{\overline{D}}\eta(y_{1},\dots,y_{n+1}).$

The form $\eta$ represents the Beilinson’s regulator evaluated in the Minor
symbol $\\{y_{1},\dots,y_{n+1}\\}$. Beilinson’s conjecture, that we will not
describe here, together with Theorem 7 suggest that the Mahler measure
$\mathrm{m}(F)$ should be related to an $L$-function associated to
$Z_{F}^{\mathrm{reg}}$ when the boundary of the Deninger cycle is trivial.
There are several technicalities that need to be taken into consideration (for
example, the fact that $Z_{F}^{\mathrm{reg}}$ is not projective), that we will
not discuss here.

A special case arises when $\eta(y_{1},\dots,y_{n+1})$ is an exact form and
the boundary of the Deninger cycle $\partial\overline{D}$ is non-trivial.
Writing $\eta(y_{1},\dots,y_{n+1})=\mathrm{d}\,\omega(y_{1},\dots,y_{n+1})$,
Stokes’s formula gives

$\mathrm{m}(F)=\mathrm{m}(c_{k})+\frac{(-1)^{n}}{(2\pi
i)^{n}}\int_{\partial\overline{D}}\omega(y_{1},\dots,y_{n+1}),$

where $\partial\overline{D}$ is the oriented boundary of the Deninger cycle.
As proposed by Maillot [Mai03] after an idea of Darboux [Dar75],
$\partial\overline{D}$ is contained in the algebraic subvariety

$W_{F}:F(y_{1},\dots,y_{n+1})=\overline{F}\left(\frac{1}{y_{1}},\dots,\frac{1}{y_{n+1}}\right)=0,$

and this suggests that $\mathrm{m}(F)$ is related to an $L$-function
associated to $W_{F}$.

Under these conditions, the computation of the Mahler measure may proceed
depending on the properties of $\omega(y_{1},\dots,y_{n+1})$. A possibility is
that $\omega$ is exact, and we can further apply Stokes’s formula.

For example, in three variables, when the $1$-form $\omega$ is exact, its
primitive can be expressed in terms of a trilogarithm. This is the case of
formulas (1), (2), or the polynomials $R_{2}$ and $T_{1}$ (the polynomial
$S_{0}$ has the same Mahler measure as (1)). The Mahler measure formulas
corresponding to these polynomials result in the evaluation of a trilogarithm
in the intersection of $\\{|x|=|y|=|z|=1\\}$ with a curve $W_{F}$ of genus 0.
For example, let us look at (2). Then

$W_{F}:(x+1)(xy^{2}-y^{2}+xy+y-x+1)=0,$

the union of a line and a genus 0 curve. What is interesting about applying
Theorem 1 is that it results in evaluations of higher genus curves that still
lead to simple Mahler measure formulas. For example, let us look at (4). Then

$W_{F}:(x^{2}+x+1)(x^{2}y^{2}-y^{2}+x^{2}y+xy+y-x^{2}+1)=0,$

and the main factor is a genus 1 curve. Analogously, if we consider the
corresponding curves $W_{F}$ for formulas (5) and (6), we obtain genus $3$ and
$4$ curves respectively.

In other cases, such as in formula (3), $\omega$ is not exact. This poses a
challenge to the evaluation of the integral, which explains why formula (3) is
only known numerically, but it has not been proven. In this case, the
integration takes place in a curve $W_{F}$ of genus 1, which corresponds,
precisely, to the elliptic curve $21a1$. Here Theorem 1 results in evaluations
of higher genus curves. For example, if we consider the corresponding curves
$W_{F}$ for formulas (10), (11), and (12), we obtain genus $3,7$, and $9$
respectively.

In conclusion, Theorem 1 leads to polynomials with substantially more
complicated geometry than the original polynomials but with the same Mahler
measure. This process serves to generate highly non-trivial identities.

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Shohei<EMAIL_ADDRESS>Hao Wangconan1024hao@akane.waseda.jp2,1
Yoshitaka<EMAIL_ADDRESS>OMRON SINIC X Corporation
Nagase Hongo Building 3F, 5-24-5 Hongo,
Bunkyo-ku, Tokyo, Japan Waseda University
1-6-1 Nishi-Waseda, Shinjuku-ku,
Tokyo, Japan SciPostLayout

# SciPostLayout: A Dataset for Layout Analysis and Layout Generation of
Scientific Posters

###### Abstract

Scientific posters are used to present the contributions of scientific papers
effectively in a graphical format. However, creating a well-designed poster
that efficiently summarizes the core of a paper is both labor-intensive and
time-consuming. A system that can automatically generate well-designed posters
from scientific papers would reduce the workload of authors and help readers
understand the outline of the paper visually. Despite the demand for poster
generation systems, only a limited research has been conduced due to the lack
of publicly available datasets. Thus, in this study, we built the
SciPostLayout dataset, which consists of 7,855 scientific posters and manual
layout annotations for layout analysis and generation. SciPostLayout also
contains 100 scientific papers paired with the posters. All of the posters and
papers in our dataset are under the CC-BY license and are publicly available.
As benchmark tests for the collected dataset, we conducted experiments for
layout analysis and generation utilizing existing computer vision models and
found that both layout analysis and generation of posters using SciPostLayout
are more challenging than with scientific papers. We also conducted
experiments on generating layouts from scientific papers to demonstrate the
potential of utilizing LLM as a scientific poster generation system. The
dataset is publicly available at https://huggingface.co/datasets/omron-
sinicx/scipostlayout_v2. The code is also publicly available at
https://github.com/omron-sinicx/scipostlayout.

##  1 Introduction

Figure 1: Example posters and annotations in SciPostLayout.

Scientific posters are used to efficiently present the contributions of a
scientific paper in a graphical format. A well-designed scientific poster
conveys the essential elements of the research and requires less time to read
than a paper. Unfortunately, creating a scientific poster that efficiently
summarizes a paper is both labor-intensive and time-consuming. Although
automating this task by utilizing ML models has shown promise, research on
scientific poster generation remains scarce due to the high complexity and
multimodality of the task. Previous studies [Qiang et al.(2019)Qiang, Fu, Yu,
Guo, Zhou, and Sigal, Xu and Wan(2021)] have built datasets to evaluate poster
generation systems for scientific papers, but these datasets are either not
publicly available or the data license is unclear, leaving this research field
lacking in gold-standard benchmarks.

Previous studies [Zhong et al.(2019)Zhong, Tang, and Yepes, Deka et
al.(2017)Deka, Huang, Franzen, Hibschman, Afergan, Li, Nichols, and Kumar]
have published publicly available datasets for layout analysis [Gu et
al.(2021)Gu, Kuen, Morariu, Zhao, Jain, Barmpalios, Nenkova, and Sun, Huang et
al.(2022)Huang, Lv, Cui, Lu, and Wei, Li et al.(2022)Li, Xu, Lv, Cui, Zhang,
and Wei, Zhong et al.(2019)Zhong, Tang, and Yepes] and layout generation [Deka
et al.(2017)Deka, Huang, Franzen, Hibschman, Afergan, Li, Nichols, and Kumar,
Fu et al.(2022)Fu, Wang, McDuff, and Song, Kikuchi et al.(2021a)Kikuchi,
Otani, Yamaguchi, and Simo-Serra, Yamaguchi(2021), Zheng et al.(2019)Zheng,
Qiao, Cao, and Lau]. However, these datasets focus on scientific papers or
mobile application design, and neglect to examine the layout of scientific
poster. Layout analysis and layout generation are important tasks for
identifying the characteristics of effective layout design in a particular
domain. Developing layout analysis models would be helpful in terms of
identifying layout patterns common to well-designed posters. In addition, by
developing layout generation models, we can generate layouts that follow the
revealed patterns of attractive posters. In other words, building a layout
dataset of scientific posters that can be utilized for these tasks would
contribute significantly to the research on scientific poster generation.

In this paper, we introduce SciPostLayout, the first scientific poster layout
dataset for layout analysis and generation. This dataset includes 7,855
scientific posters with manual layout annotations. All posters included in the
dataset are under the CC-BY license
111https://creativecommons.org/licenses/by/4.0/. Note that some of the older
posters are under older licenses, such as CC-BY 2.0 and CC-BY 3.0.. Figure 1
shows examples of posters and annotations of SciPostLayout. SciPostLayout can
be used to evaluate layout analysis and generation systems in the same way as
existing datasets. Both the layout analysis and generation on SciPostLayout
are more challenging than scientific papers because of the various positions
of elements such as figures and tables. In addition, we manually collected 100
papers associated with posters in the dataset to utilize SciPostLayout in
layout generation from scientific papers. These papers are also under the CC-
BY license.

We evaluated existing models for layout analysis and generation on
SciPostLayout. In the layout analysis task, we found that although the
existing models can recognize certain elements with high accuracy, they are
less effective than on the scientific paper dataset, indicating that layout
analysis in SciPostLayout is more complicated. In the layout generation task,
an LLM-based model can generate aligned layouts with less overlap, but we
found that layout generation on SciPostLayout is also a more challenging task
compared to generation on the scientific paper dataset. In addition, we
implemented LLM-based models for generating poster layouts from papers and
discussed the potential of poster generation utilizing LLM.

##  2 Related Work

In this section, we describe related work on layout analysis, layout
generation, and scientific poster generation.

###  2.1 Layout Analysis

Layout analysis is a task in which a system detects the layouts of
unstructured documents by predicting bounding boxes and categories such as
tables or figures. This task is generally categorized as an object detection
problem [Gu et al.(2021)Gu, Kuen, Morariu, Zhao, Jain, Barmpalios, Nenkova,
and Sun, Li et al.(2022)Li, Xu, Lv, Cui, Zhang, and Wei, Zhong et
al.(2019)Zhong, Tang, and Yepes].

The PubLayNet dataset [Zhong et al.(2019)Zhong, Tang, and Yepes] is one of the
well-known layout analysis datasets. It contains scientific paper images
automatically annotated with bounding boxes and polygonal segmentation across
five categories: text, title, list, figure, and table. Previous studies have
demonstrated that deep neural networks, including UDoc [Gu et al.(2021)Gu,
Kuen, Morariu, Zhao, Jain, Barmpalios, Nenkova, and Sun], DiT [Li et
al.(2022)Li, Xu, Lv, Cui, Zhang, and Wei], and LayoutLMv3 [Huang et
al.(2022)Huang, Lv, Cui, Lu, and Wei], which are trained on the PubLayNet
dataset, can recognize the layouts of scientific papers with high accuracy.
These models utilize Faster R-CNN [Ren et al.(2015)Ren, He, Girshick, and
Sun], Mask R-CNN [He et al.(2017)He, Gkioxari, Dollár, and Girshick], and
Cascade R-CNN [Cai and Vasconcelos(2018)] as object detection models and
ResNet [He et al.(2016)He, Zhang, Ren, and Sun] and Vision Transformer
[Dosovitskiy et al.(2020)Dosovitskiy, Beyer, Kolesnikov, Weissenborn, Zhai,
Unterthiner, Dehghani, Minderer, Heigold, Gelly, et al.] as visual backbones.

Other studies have also proposed layout analysis datasets, such as Article
Regions [Soto and Yoo(2019)], which is a dataset of region-annotated
scientific papers from PubMed Central, TableBank [Li et al.(2019b)Li, Cui,
Huang, Wei, Zhou, and Li], an image-based table detection dataset built with
Word and LaTeX documents, and DocBank [Li et al.(2020)Li, Xu, Cui, Huang, Wei,
Li, and Zhou], a document-level dataset with token annotations.

In contrast to the previous datasets, SciPostLayout focuses on scientific
posters. The layout analysis of scientific posters is more challenging than
that of scientific papers because of the variety of fonts and positions of
figures and tables.

###  2.2 Layout Generation

Layout generation is a task that arose from the needs of design applications,
including magazine covers, UI interfaces, presentation slides, and banner
advertising [Deka et al.(2017)Deka, Huang, Franzen, Hibschman, Afergan, Li,
Nichols, and Kumar, Fu et al.(2022)Fu, Wang, McDuff, and Song, Kikuchi et
al.(2021a)Kikuchi, Otani, Yamaguchi, and Simo-Serra, Yamaguchi(2021), Zheng et
al.(2019)Zheng, Qiao, Cao, and Lau]. Layout generation can be categorized into
unconditional generation [Arroyo et al.(2021)Arroyo, Postels, and Tombari,
Gupta et al.(2021)Gupta, Lazarow, Achille, Davis, Mahadevan, and Shrivastava,
Jiang et al.(2022)Jiang, Sun, Zhu, Lou, and Zhang], which generates layouts
without any constraints from the user, and conditional generation, which
enables the user to create their own desired layout with constraints. There
are also several subcategories within conditional generation, each giving the
system different constraints for generation, such element types [Kikuchi et
al.(2021b)Kikuchi, Simo-Serra, Otani, and Yamaguchi, Kong et al.(2022)Kong,
Jiang, Chang, Zhang, Hao, Gong, and Essa, Lee et al.(2020)Lee, Jiang, Essa,
Le, Gong, Yang, and Yang], element types and sizes [Kong et al.(2022)Kong,
Jiang, Chang, Zhang, Hao, Gong, and Essa], relationships between elements
[Kikuchi et al.(2021b)Kikuchi, Simo-Serra, Otani, and Yamaguchi, Lee et
al.(2020)Lee, Jiang, Essa, Le, Gong, Yang, and Yang], completion [Gupta et
al.(2021)Gupta, Lazarow, Achille, Davis, Mahadevan, and Shrivastava], and
refinement [Rahman et al.(2021)Rahman, Sermuga Pandian, and Jarke].

Early studies explored generative adversarial networks (GANs) [Kikuchi et
al.(2021b)Kikuchi, Simo-Serra, Otani, and Yamaguchi, Li et al.(2019a)Li, Yang,
Hertzmann, Zhang, and Xu] and variational autoencoders (VAEs) [Arroyo et
al.(2021)Arroyo, Postels, and Tombari, Jiang et al.(2022)Jiang, Sun, Zhu, Lou,
and Zhang, Yamaguchi(2021)] for layout generation. Motivated by the success of
the Transformer architecture [Vaswani et al.(2017)Vaswani, Shazeer, Parmar,
Uszkoreit, Jones, Gomez, Kaiser, and Polosukhin] in NLP, some studies treated
layout generation as a sequence-to-sequence problem [Gupta et al.(2021)Gupta,
Lazarow, Achille, Davis, Mahadevan, and Shrivastava, Jiang et al.(2023)Jiang,
Guo, Sun, Deng, Wu, Mijovic, Yang, Lou, and Zhang, Kong et al.(2022)Kong,
Jiang, Chang, Zhang, Hao, Gong, and Essa, Lin et al.(2023)Lin, Guo, Sun, Yang,
Lou, and Zhang]. Moreover, discrete diffusion models have achieved notable
results [Chai et al.(2023)Chai, Zhuang, and Yan, Hui et al.(2023)Hui, Zhang,
Zhang, Xie, Wang, and Lu, Inoue et al.(2023)Inoue, Kikuchi, Simo-Serra, Otani,
and Yamaguchi, Zhang et al.(2023)Zhang, Guo, Sun, Lou, and Zhang]. Datasets
including Rico [Deka et al.(2017)Deka, Huang, Franzen, Hibschman, Afergan, Li,
Nichols, and Kumar], Magazine [Zheng et al.(2019)Zheng, Qiao, Cao, and Lau],
and PubLayNet [Zhong et al.(2019)Zhong, Tang, and Yepes] are typically
utilized to evaluate layout generation models.

We evaluate existing models for layout generation in the scientific poster
domain. Generating layouts for scientific posters is challenging, as are other
layout generation tasks that require the positioning of figures and tables.

###  2.3 Scientific Poster Generation

As stated earlier, scientific poster generation is a task in which a system
generates a poster summarizing the important information in a scientific
paper. Although this task is promising for applications, research on
scientific poster generation remains scarce due to its high complexity and
multimodality.

Paramita and Khodra [Paramita and Khodra(2016)] treated scientific poster
generation as a summarization problem and developed a model to extract
essential sentences from papers by classification and fill them into templates
to generate posters. However, the generated posters are composed only of
texts, making them unsuitable for efficiently conveying information.

Qiang et al. [Qiang et al.(2019)Qiang, Fu, Yu, Guo, Zhou, and Sigal]
decomposed poster generation into three subtasks: content extraction, panel
attribute inference, and panel layout generation. For content extraction,
their system uses TextRank [Mihalcea and Tarau(2004)] to summarize each
section of a paper and then manually pick the important figures and tables.
For panel attributes inference, the system uses a Bayesian network trained to
learn the size and aspect ratio of each panel in a poster. For panel layout
generation, the system generates panel layouts with a binary tree structure
recursively. Finally, with the inferred data, posters are generated in a LaTeX
format. The authors also constructed a dataset with 85 paper-poster pairs and
facilitated evaluations of the model performances by experts on three metrics:
readability, informativeness, and aesthetics. However, the licensing status of
the posters is unclear, and the layout annotation is not detailed enough; for
example, both figures and tables are simply labeled as Figure.

Xu and Wan [Xu and Wan(2021)] focused on the content extraction part of poster
generation and proposed a model to extract text, figures, and tables from
papers simultaneously. They additionally proposed a three-step framework to
generate posters: section filter, content extraction, and poster composition.
They also built a dataset for evaluation, which is currently awaiting public
release.

Our SciPostLayout dataset is the first to contain scientific posters and
papers with the CC-BY license, which can be used to evaluate scientific poster
generation systems.

##  3 Dataset

Previous studies [Qiang et al.(2019)Qiang, Fu, Yu, Guo, Zhou, and Sigal, Xu
and Wan(2021)] have built datasets to evaluate poster generation systems for
scientific papers. However, these datasets are either not publicly available
or the data licensing is unclear. In this paper, we construct SciPostLayout, a
fully public dataset with all data under the Creative Commons license, which
allows unrestricted dissemination, adaptation, and re-use.

First, we downloaded posters in PDF format from F1000Research
222https://f1000research.com/browse/posters. Among these, we retained 7,943
posters under the CC-BY license. Posters under non-distributable and
noncommercial licenses were excluded. The PDF files were converted into PNG
format at DPI=100 for the following annotations. We excluded posters with file
sizes below 200KB as they mainly consisted of text, which is unsuitable for
layout analysis. After this exclusion, 7,855 posters remained. We investigated
the word trends in the titles and found that most of the collected posters
were in the biomedical field.

Next, we recruited professional data annotators to manually annotate the
document layout of the 7,855 posters. The layouts of scientific posters are
more diverse than the layouts of the papers in PubLayNet [Zhong et
al.(2019)Zhong, Tang, and Yepes] because of the variety of fonts, size and
position of figures and tables, and typography. We expanded PubLayNet’s five-
category annotation standard to nine categories to acquire fine-grained
annotations of the layouts. The annotation criteria are listed in Table 1.

Category | Contents
---|---
Title | paper title
Author Info | author, author affiliation
Section |  section titlea
Text |  paragraphb
List |  nested listc
Table | main body of table
Figure |  main body of figured
Caption | caption of table and figure
Unknown |  advertising information, logo of affiliatione

aOnly the highest level of sections are annotated as Section. Subsections are
annotated as Text because of the small font size.
bWhen multiple paragraphs are connected, they are annotated as a single
object. When a subsection is inserted between paragraphs, a new bounding box
is created. The subsection name is not enclosed in a separate bounding box,
but is included together with the subsequent paragraph. Footnotes are ignored
during annotation.
cNot only bullet points ($\cdot$) but also paragraph numbers (e.g. 1., a.) are
treated as lists. It includes reference block.
dWhen sub-figures exist, the whole figure panel is annotated as a single
object.
eOnly elements with big areas are annotated.

Table 1: Annotation criteria of SciPostLayout.

Subsequently, we manually searched for the papers paired with the posters
because there is no information relating to the papers on F1000Research. We
found 100 papers under the CC-BY license and then divided these 100 paper-
poster pairs into dev/test sets comprising 50 pairs each.

After the above processes, we obtained our dataset SciPostLayout, which
includes train data with 6,855 posters and manual layout annotations as well
as dev/test data with 500 posters, 50 papers paired with partial posters, and
manual layout annotations. Table 2 lists the statistics of the dataset.
SciPostLayout is the first scientific poster layout dataset that can be used
for evaluating layout analysis and layout generation systems. Moreover,
SciPostLayout is the first dataset that contains scientific paper-poster pairs
with the CC-BY license, which can be utilized for evaluating scientific poster
generation systems.

Split | Title | Author Info | Section | Text | List | Table | Figure | Caption | Unknown | Total
---|---|---|---|---|---|---|---|---|---|---
Train | 6,847 | 6,709 | 36,083 | 46,297 | 20,723 | 4,916 | 33,665 | 13,065 | 8 | 168,313
Dev | 497 | 492 | 2,609 | 3,215 | 1,489 | 401 | 2,421 | 1,015 | 2 | 12,141
Test | 498 | 494 | 2,646 | 3,199 | 1,583 | 328 | 2,484 | 1,010 | 1 | 12,243

Table 2: Statistics of train, dev, and test data in SciPostLayout.

##  4 Experiments

We conducted three experiments using the collected dataset: layout analysis,
layout generation, and paper-to-layout. We compared existing models for each
experiment.

###  4.1 Layout Analysis

We used LayoutLMv3 [Huang et al.(2022)Huang, Lv, Cui, Lu, and Wei] and DiT [Li
et al.(2022)Li, Xu, Lv, Cui, Zhang, and Wei] for the layout analysis task. We
started from the base size checkpoints with Cascade R-CNN detectors and fine-
tuned them on SciPostLayout train data for both models. The checkpoint with
the highest performances on the dev set was used for evaluation.

We measured the performance using the mean average precision (mAP) @
intersection over union (IoU) [0.50:0.95] of bounding boxes, the results of
which are reported in Table 3. The Unknown category was omitted due to an
insufficient number of elements. LayoutLMv3 outperformed DiT in all
categories. In addition, both models showed high performances in the Title and
Author Info categories. We attribute this result to the regularity of the
Title and Author Info blocks, since they are always at the top of the posters
and there is usually only one of each block per poster. However, compared to
the results on PubLayNet [Huang et al.(2022)Huang, Lv, Cui, Lu, and Wei, Li et
al.(2022)Li, Xu, Lv, Cui, Zhang, and Wei], in which mAP@IoU is over 90, both
models showed a performance drop, indicating the complexity of our dataset.

Model | Title | Author Info | Section | Text | List | Table | Figure | Caption | Overall
---|---|---|---|---|---|---|---|---|---
LayoutLMv3 | 88.32 | 85.79 | 71.98 | 71.33 | 73.71 | 74.41 | 67.60 | 69.24 | 66.93
DiT | 85.23 | 79.89 | 70.64 | 67.66 | 69.93 | 74.16 | 67.12 | 59.27 | 63.77

Table 3: Layout analysis performance (mAP@IoU[0.50:0.95]) on SciPostLayout
test set. Bold numbers indicate highest performances.

###  4.2 Layout Generation

We conducted layout generation experiments with various settings for the
information to be input into the model [Jiang et al.(2023)Jiang, Guo, Sun,
Deng, Wu, Mijovic, Yang, Lou, and Zhang]. Generation conditioned on types
(Gen-T) aims to generate layouts from the number of element types
(categories). Generation conditioned on types and sizes (Gen-TS) aims to
generate layouts from the number and size of element types. Generation
conditioned on relationships (Gen-R) aims to generate layouts from the number
of element types and position relationships between the elements. Completion
means generating a complete layout from a part of the layout. Refinement means
generating a new layout from a layout that needs improvement.

We used LayoutDM [Inoue et al.(2023)Inoue, Kikuchi, Simo-Serra, Otani, and
Yamaguchi], LayoutFormer++ [Jiang et al.(2023)Jiang, Guo, Sun, Deng, Wu,
Mijovic, Yang, Lou, and Zhang], and LayoutPrompter [Lin et al.(2023)Lin, Guo,
Sun, Yang, Lou, and Zhang] for the layout generation task. Unlike the layout
analysis models, the layout generation models were trained from randomly
initialized parameters. We did not train LayoutPrompter because it uses
GPT-4333gpt-4-1106-preview via API.

We evaluated the model performances using maximum IoU (mIoU), Alignment,
Overlap, and Fréchet Inception Distance (FID). mIoU is a measure of the
highest IoU between a generated layout and a real layout [Kikuchi et
al.(2021b)Kikuchi, Simo-Serra, Otani, and Yamaguchi, Lin et al.(2023)Lin, Guo,
Sun, Yang, Lou, and Zhang]. Alignment indicates how well the elements in a
layout are aligned with each other [Li et al.(2021)Li, Yang, Zhang, Liu, Wang,
and Xu, Lin et al.(2023)Lin, Guo, Sun, Yang, Lou, and Zhang]. Overlap is the
overlapping area between two arbitrary elements in a layout [Li et
al.(2021)Li, Yang, Zhang, Liu, Wang, and Xu, Lin et al.(2023)Lin, Guo, Sun,
Yang, Lou, and Zhang]. FID measures how similar the distribution of the
generated layouts is to that of real layouts [Kikuchi et al.(2021b)Kikuchi,
Simo-Serra, Otani, and Yamaguchi, Lin et al.(2023)Lin, Guo, Sun, Yang, Lou,
and Zhang]. A higher mIoU value means higher performance; for the other
metrics, a lower value means higher performance. Note that both mIoU and FID
are evaluation metrics based on similarity to the real layouts, but mIoU is
calculated from the intersection between layouts, while FID is an embedding-
based evaluation metric. Thus, the hierarchical order of mIoU and FID
performances across models is not always consistent.

The results are shown in Table 4. mIoU was low for all models and was less
than half that in PubLayNet [Inoue et al.(2023)Inoue, Kikuchi, Simo-Serra,
Otani, and Yamaguchi, Jiang et al.(2023)Jiang, Guo, Sun, Deng, Wu, Mijovic,
Yang, Lou, and Zhang, Lin et al.(2023)Lin, Guo, Sun, Yang, Lou, and Zhang]. In
contrast, all models performed effectively on Alignment, indicating that they
can generate aligned layouts. LayoutPrompter was the most effective for
Overlap, indicating that it generates layouts with the least overlap. LayoutDM
was the most effective in terms of FID, indicating that it can generate
layouts most similar to real layout distribution on each setting. Comparing
the FID and Overlap of LayoutDM and LayoutPrompter shows that LayoutDM tends
to generate layouts that are similar to the real layout distribution but with
more overlap, while LayoutPrompter generates layouts with less overlap but
that are farther from the real layout distribution. In the Refinement setting,
LayoutPrompter outperformed the other models, indicating that it can generate
layouts similar to real layouts from noisy layouts. Figure 2 shows examples of
generated layouts of each model and the real layout in the Refinement setting.
LayoutPrompter generated the most similar layout to the real layout, while
LayoutDM and LayoutFormer++ generated layouts with overlap that were not
similar to the real layout.

Task | Model |  mIoU$\uparrow$ |  Alignment$\downarrow$ |  Overlap$\downarrow$ |  FID$\downarrow$
---|---|---|---|---|---
| LayoutDM | 0.073 | 0.002 | 0.524 | 1.524
Gen-T | LayoutFormer++ | 0.079 | 0.002 | 0.242 | 26.999
| LayoutPrompter | 0.084 | 0.000 | 0.231 | 9.470
| LayoutDM | 0.113 | 0.002 | 0.564 | 1.419
Gen-TS | LayoutFormer++ | 0.089 | 0.003 | 0.245 | 27.372
| LayoutPrompter | 0.102 | 0.000 | 0.182 | 8.605
| LayoutDM | 0.075 | 0.001 | 0.571 | 5.556
Gen-R | LayoutFormer++ | 0.066 | 0.002 | 0.553 | 20.296
| LayoutPrompter | 0.069 | 0.000 | 0.317 | 6.282
| LayoutDM | – | 0.001 | 0.610 | 2.490
Completion | LayoutFormer++ | – | 0.001 | 0.250 | 5.840
| LayoutPrompter | – | 0.000 | 0.011 | 10.874
| LayoutDM | 0.119 | 0.002 | 0.531 | 1.196
Refinement | LayoutFormer++ | 0.301 | 0.002 | 0.289 | 10.955
| LayoutPrompter | 0.552 | 0.001 | 0.112 | 0.152
Table 4: Layout generation performance on SciPostLayout test set. Bold numbers
indicate the highest performances. $\uparrow$ indicates larger values are
ideal, $\downarrow$ indicates smaller values are ideal. mIoU is omitted from
the Completion setting because the number of elements of generated layouts
differs from that of real layouts. Figure 2: Examples of generated layouts and
the real layout in the Refinement setting. Examples of the other settings are
provided in Appendix A.

###  4.3 Paper-to-Layout

In Section 4.2, we conducted experiments on layout generation from real
constraints. To develop a system that automatically generates a poster from a
scientific paper, we need to implement a model that extracts constraints or
generates a layout from a paper. Thus, we implement and compare models to
generate layouts from scientific papers in two settings: Gen-T and Gen-P.
Figure 3 illustrates the settings of Gen-T and Gen-P. In the Gen-T setting,
GPT-4 first extracts element type constraints from a scientific paper. Then, a
layout generation model generates a layout from the extracted constraints, the
same as the Gen-T setting in Section 4.2. In the Gen-P setting, GPT-4 first
generates a summary within 1,000 words from a scientific paper. Then,
LayoutPrompter with GPT-4 generates a layout from the paper summary.
LayoutPrompter does not directly generate a layout from a paper because the
entire paper is too long for document retrieval when it searches for examples
similar to the target paper for layout generation, which are given as few-shot
examples. We used 50 paper-poster pairs of dev set as few-shot examples in the
Gen-P setting. Different from Section 4.1 and 4.2, we used only 50 paper-
poster pairs of test set to evaluate the model performances. In both settings,
Nougat [Blecher et al.(2023)Blecher, Cucurull, Scialom, and Stojnic] was
utilized to extract texts from a paper in PDF format. The prompts to extract
element type constrains or generate a paper summary are shown in Appendix B.

Figure 3: Gen-T and Gen-P settings of Paper-to-Layout.

We evaluated the accuracy of extracting element type constraints in the Gen-T
setting based on mean absolute error (MAE). MAE is calculated as the mean of
the absolute differences between the number of elements predicted by GPT-4 and
the actual number of elements in the poster layout. For instance, if GPT-4
predicts 4 Figures while the actual poster has 5, the absolute error for this
element type would be |4-5| = 1. We performed extraction three times and
calculated the mean value. Table 5 shows the results. For Title and Author
Info, the generated number of elements is close to the real constraints. For
Text and Figure, GPT-4 shows large error values because the number of these
elements is larger than the number of other elements and there is also a large
range of values, as shown in Table 2. However, the overall MAE is less than 2,
indicating that GPT-4 can generate constraints close to the real number of
elements.

Model | Title | Author Info | Section | Text | List | Table | Figure | Caption | Overall
---|---|---|---|---|---|---|---|---|---
GPT-4 | 0.06 | 0.04 | 1.90 | 4.9 | 2.31 | 0.57 | 3.66 | 2.43 | 1.98
Table 5: MAE results of element type extraction by GPT-4.

We conducted experiments on poster layout generation from scientific papers in
the Gen-T and Gen-P settings, which are reported in Table 6. The number of
each element of the extracted constraints in the Gen-T setting was given to
the models as the mean value rounded down to the nearest whole number. All
models performed effectively on Alignment, indicating that they can generate
aligned layouts in the same way as layout generation from real constraints in
Section 4.2. LayoutPrompter in the Gen-P setting was the most effective for
Overlap, indicating that it generates layouts with the least overlap.
Comparing the FID values in the Gen-T setting to the values in the Gen-T
setting of Table 4, we find that the FID values of all models slightly
increase, indicating that generating layouts similar to real layouts from
extracted constraints is more difficult than generating layouts from real
constraints. Figure 4 shows examples of layouts generated by LayoutPrompter in
the Gen-T and Gen-P settings. Although the generated layouts are not similar
to the real layout, LayoutPrompter could generate aligned layouts with less
overlap. The experimental results in this section demonstrate that although
the current LLM has difficulty generating layouts similar to real layouts from
scientific papers, it can generate high-quality layouts, indicating that LLM
has potential as a poster generation system from scientific papers.

Task | Model |  Alignment$\downarrow$ |  Overlap$\downarrow$ |  FID$\downarrow$
---|---|---|---|---
| LayoutDM | 0.001 | 0.542 | 9.073
Gen-T | LayoutFormer++ | 0.001 | 0.617 | 56.396
| LayoutPrompter | 0.000 | 0.256 | 19.892
Gen-P | LayoutPrompter | 0.000 | 0.022 | 17.790
Table 6: Paper-to-Layout generation performance on SciPostLayout test set.
Bold numbers indicate the highest performances. $\downarrow$ indicates smaller
values are ideal. mIoU is omitted because the number of elements of generated
layouts differs from that of real layouts. Figure 4: Example of LayoutPrompter
generated layout and real layout in the Gen-T and Gen-P setting from
scientific paper.

##  5 Conclusion

In this paper, we built a new dataset called SciPostLayout, which consists of
7,855 scientific posters and 100 scientific papers downloaded from a website.
All posters in SciPostLayout are manually annotated with categories of layout
elements (such as titles and figures) to be used in layout analysis and
generation tasks. In addition, SciPostLayout is available for commercial
research because all of the posters and papers are under the CC-BY license. We
conducted layout analysis and generation experiments to evaluate the
performances of existing models on SciPostLayout. For layout analysis,
although some elements could be recognized with high accuracy, we found that
layout analysis on SciPostLayout was more difficult than on a scientific paper
dataset. For layout generation, although existing models could generate
aligned layouts, we found it was difficult to generate layouts that are
similar to real layouts. In addition, we implemented and evaluated GPT-4-based
models to generate poster layouts from scientific papers. The experimental
results proved that LLM has potential as a scientific poster generation
system. Our future work will involve developing a model to improve layout
analysis and generation. In addition, we will investigate improved methods for
generating poster layouts and extracting contents from scientific papers.

## Acknowledgements

This work was supported by JST Moonshot R&D Program, Grant Number JPMJMS2236.
We would like to thank Naoto Inoue, Ph. D. of CyberAgent Inc. AILab for useful
comments about layout generation models.

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