On Salem numbers which are exceptional units

Toufik Zaïmi

Abstract. By extending a construction due to Gross and McMullen [6], we show
that for any odd integer $n$ and for any even integer $d\geq n+3$ there are
infinitely many Salem numbers $\alpha$ of degree $d$ such that $\alpha^{n}-1$
is a unit. A similar result is proved when $n$ runs through some classes of
even integers, $d\geq n+4$ and $d/2$ is an odd integer.

2010 MSC: 11R06, 11R04, 11Y40.

Key words and phrases: Salem numbers, exceptional units, unramified Salem
numbers.

1\. Introduction

A Salem number is a real algebraic integer greater than $1,$ whose other
conjugates lie inside the closed unit disc, with at least one conjugate lying
on the boundary; the set of such numbers is traditionally denoted by
$\mathbb{T\ }$[2].

Let $S_{\alpha}=S$ designate the minimal polynomial of a Salem number
$\alpha.$ Then, $S$ has two real roots, namely $\alpha_{1}:=\alpha$ and
$\alpha^{-1},$ and the rest, say $\alpha_{2}^{\pm 1},...,\alpha_{t}^{\pm 1},$
lie on the unit circle. Also, $\deg(S)=2t\geq 4,$ $S$ is reciprocal, i. e.,
$S(x)=x^{t}S(1/x),$ and the polynomial
$T_{\alpha}(x)=T(x):=(x-(\alpha_{1}+\alpha_{1}^{-1}))(x-(\alpha_{2}+\overline{\alpha_{2}}))\cdot\cdot\cdot(x-(\alpha_{t}+\overline{\alpha_{t}}))\in\mathbb{Z}[x],$
called the trace polynomial of $S$ (or the trace polynomial of $\alpha)$
satisfies

$S(x)=x^{t}T(x+\frac{1}{x}).$ (1)

Clearly, $T_{\alpha}$ is irreducible and has one root greater than $2,$ namely
$\alpha+\alpha^{-1}$ (called in [6] a Salem trace number), and $(t-1)\geq 1$
roots belonging to the interval $(-2,2).$ Conversely, an algebraic integer
$\beta>2$ of degree $t\geq 2$ whose other conjugates lie in $(-2,2)$ is a
Salem trace number associated to a Salem number $\alpha$ of degree $2t,$ via
the relation $\beta=\alpha+\alpha^{-1}.$ Some properties of Salem trace
polynomials may be found in [10].

An algebraic integer $u$ is said to be an exceptional unit if both $u$ and
$u-1$ are units [7]. Siegel proved that there are at most a finite number of
exceptional units in any number field $F,$ and a result of Evertse [5] implies
that this number is no more than $3(7^{3\deg(F)}).$

Clearly, a Salem number is a unit. Theorem 3 of [8] yields that every integer
greater than $2$ is a limit of a sequence of Salem numbers which are
exceptional units, and a characterization of Salem numbers $\alpha$ that are
exceptional units among the elements of the set $\mathbb{T},$ in terms of the
fractional parts of the powers of $\alpha,$ follows from the main result of
[9].

In [7], Silverman conducted some investigations suggesting that if a unit $u$
has small Mahler measure (the Mahler measure of an algebraic integer is the
absolute value of the product of its conjugates of modulus greater than $1),$
then there are many values of $n$ for which $u^{n}-1$ is a unit. In
particular, he proved that the numbers of such values of $n$ is bounded above
by $O(d^{1+0.7/\log\log d}),$ where $d=\deg(u).$ In most of his numerical
investigations Silverman supposed that $u$ is a Salem number from Boyd’s lists
[3-4], and one of his observations says that among the forty two known small
Salem numbers thirty three are exceptional units, and there is a Salem number
$\alpha$ of degree $26$ such that $\alpha^{n}-1$ is a unit for all
$n\in\\{1,2,...,10\\}.$ From Theorem 2, below, we can deduce that for any
finite set $F$ of natural numbers there is an infinite subset $N$ of
$2\mathbb{N}$ such that for each $d\in N$ there exist infinitely many Salem
numbers $\alpha$ of degree $d$ having the property that $\alpha^{n}-1$ is a
unit for all $n\in F.$

By considering the problem of the realization of a monic irreducible
polynomial with integer coefficients as the characteristic polynomial of an
automorphism of the even indefinite unimodular lattice, Gross and McMullen [6]
were concerned with the elements $\alpha$ of $\mathbb{T}$ such that
$\alpha^{2}-1$ is a unit. They called such numbers $\alpha$ unramified Salem
numbers, and they showed in this case that $\deg(\alpha)/2$ must be odd [6,
Proposition 3.3]. Also, they gave a construction leading to the fact that for
any odd integer $t\geq 3$ there are infinitely many unramified Salem numbers
of degree $2t.$

Since a power of a unit (resp. of a Salem number of degree $2t$) is a unit
(resp. is a Salem number of degree $2t$), a Salem number is unramified if and
only if its square is an exceptional unit, and a natural question arises
immediately:

For which integers $n\geq 1$ and $t\geq 2$ are there infinitely many Salem
numbers $\alpha$ of degree $2t$ such that $\alpha^{n}-1$ is a unit?

The aim of the present paper is to give some partial answers to this question.

Theorem 1. There exist infinitely many Salem numbers $\alpha$ of degree $2t$
such that $\alpha^{n}-1$ is a unit, whenever one of the following conditions
holds:

(i) $n$ is odd and $t\geq(n+3)/2;$

(ii) $n\equiv 2\mathop{\mathrm{m}od}4,$ $t$ is odd and $t\geq(n+4)/2;$

(iii) $n=2^{s}$ for some integer $s\geq 2,$ $t$ is odd and $t\geq(n+6)/2;$

(iv) $n\equiv 4\mathop{\mathrm{m}od}8,$ $n\neq 0\mathop{\mathrm{m}od}3,$ $t$
is odd and $t\geq(n+6)/2.$

Because the degree $t$ of the polynomial trace is at least $2,$ Theorem 1(i)
is sharp when $n=1.$ According to [6, Proposition 3], Theorem 1(ii) is also
sharp for $n=2.$ As mentioned above, [6, Proposition 3.3] says that if
$\alpha\in\mathbb{T}$ and $\alpha^{n}-1$ is a unit for some even natural
natural $n,$ then $t$ is necessarily odd, since
$\alpha^{n}-1=(\alpha^{2}-1)((\alpha^{2})^{n/2-1}+(\alpha^{2})^{n/2-2}+\cdot\cdot\cdot+1).$
We will see in the next section that Theorem 1 is sharp when $n\in\\{3,4,6\\}$
but not for $n=5.$ The following result is a weak complement of Theorem 1 when
$n\equiv 0\mathop{\mathrm{m}od}4.$

Theorem 2. Let $n\in\mathbb{N}.$ Then, there is an infinite subset $N$ of
$\mathbb{N}$ such that for each $t\in N$ there are infinitely many Salem
numbers $\alpha$ satisfying $\deg(\alpha)=2t$ and $\alpha^{n}-1$ is a unit.

Given a finite sequence $n_{1}<n_{2}<\cdot\cdot\cdot<n_{s}$ of natural
numbers, if we apply Theorem 2 with $n=n_{1}n_{2}...n_{s}$ we see that there
is an infinite subset (say again) $N$ of $2\mathbb{N}$ such that for each
$d\in N$ there exist infinitely many Salem numbers $\alpha$ of degree $d$
having the property that $\alpha^{n_{k}}-1$ is a unit for each
$k\in\\{1,...s\\}.$

The first value of $n$ for which Theorem 1 does not answer our question is
$12.$ From the proof of Theorem 2, it is easy to see that for any natural
number $v\neq 0\mathop{\mathrm{m}od}3$ there are infinitely many Salem numbers
$\alpha$ of degree $18+4v$ such that $\alpha^{12}-1$ is a unit (the problem
remains open for each degree of the form $18+4v$ with $v\equiv
0\mathop{\mathrm{m}od}3).$

Throughout, when we speak about conjugates, the norm, the degree and the
minimal polynomial of an algebraic integer without mentioning the basic field,
this is meant over $\mathbb{Q}.$ By convention, an empty sum is equal to zero
and an empty product is equal to one. Also, we denote by

$C_{n}(x):=\mathop{\displaystyle\prod}\limits_{k=1}^{(n-1)/2}(x-2\cos(\frac{2k\pi}{n}))\text{
\ \ \ (resp.
}C_{n}(x):=\mathop{\displaystyle\prod}\limits_{k=1}^{(n-2)/2}(x-2\cos(\frac{2k\pi}{n})))$

the trace polynomial of $(x^{n}-1)/(x-1)$ when $n$ is odd (resp. of
$(x^{n}-1)/(x^{2}-1)$ when $n$ is even). Then, $C_{1}(x)=C_{2}(x)=1,$
$C_{n}(x)\in\mathbb{Z}[x]$ and the roots of $C_{n}$ (for $n\geq 3)$ are
distinct and belong to the interval $(-2,2).$ To prove Theorem 1 and Theorem 2
we shall use the following generalization of [6, Theorem 7.3].

Theorem 3. Let $n\geq 1$ be an odd (resp. an even) natural number, $t$ an
integer greater than or equal to $(n+3)/2$ (resp. $t$ an odd integer greater
than or equal to $(n+4)/2)$ and  $D(x)\in\mathbb{Z}[x]$ a monic separable
polynomial of degree $t-(n+3)/2$ (resp. of degree $t-(n+4)/2)$ with all roots
(if any) in $(-2,2)$ and such that $\gcd(D,C_{n})=1.$ Then, for all
sufficiently large integers $a$ the polynomial

$C_{n}(x)(x-2)D(x)(x-a)-1\text{ \ \ \ \ \ ({resp.}
}C_{n}(x)(x^{2}-4)D(x)(x-a)-1)$

is a trace polynomial (of degree $t)$ of a Salem number $\alpha$ such that
$\alpha^{n}-1$ is a unit.

The proofs of the theorems are given in the last section. In the next one we
consider Salem numbers $\alpha$ such that $\alpha^{n}-1$ is a unit for some
$n\leq 6,$ and we give a characterization of Salem polynomials $S_{\alpha}$
such that $\alpha^{n}-1$ is a unit and $n\neq 5.$ All computations are done
using the system Pari [1].

2\. On Salem numbers $\alpha$ such that $\alpha^{n}-1$ is a unit for some
$n\leq 6$

With the notation above, set

$S(x)=(1+x^{2t})+a_{1}(x+x^{2t-1})+\cdot\cdot\cdot+a_{t-1}(x^{t-1}+x^{t+1})+a_{t}x^{t},$

for the minimal polynomial of the Salem number $\alpha.$

Since the distinct numbers $\alpha_{1}^{\pm n},...,\alpha_{t}^{\pm n}$ , where
$n\in\mathbb{N},$ are the conjugates of the Salem number $\alpha^{n},$ we have
$\mathop{\mathrm{N}orm}(\alpha^{n}-1)=(\alpha_{1}^{n}-1)(\alpha_{1}^{-n}-1)(\alpha_{2}^{n}-1)(\overline{\alpha_{2}^{n}}-1)\cdot\cdot\cdot(\alpha_{t}^{n}-1)(\overline{\alpha_{t}^{n}}-1)<0$
and $\mathop{\mathrm{N}orm}(\alpha^{n}+1)>0.$ Hence, $\alpha^{n}-1$ is a unit
if and only if $\mathop{\mathrm{N}orm}(\alpha^{n}-1)=-1,$ and $\alpha^{n}+1$
is a unit if and only if $\mathop{\mathrm{N}orm}(\alpha^{n}+1)=1.$ Especially,

$\alpha-1\text{ is a unit }\Leftrightarrow S(1)=-1,$ (2)

as $S(1)=\mathop{\mathrm{N}orm}(\alpha-1),$ or equivalently

$\alpha-1\text{ is a unit}\Leftrightarrow
a_{t}=-3-2(a_{1}+\cdot\cdot\cdot+a_{t-1}).$ (3)

In a similar way we obtain a characterization of the polynomial $S$ when
$\alpha^{n}-1$ is a unit for some $n\in\\{2,3,4\\}:$

Proposition 1. With the notation above we have the following assertions.

(i) $\alpha^{2}-1$ is a unit  if and only if

$t\equiv 1\mathop{\mathrm{m}od}2,\text{ \
}a_{t}=-1-2\mathop{\displaystyle\sum}\limits_{k=0}^{(t-3)/2}a_{2k+1}\text{
{and }\ }a_{t-1}=-1-\mathop{\displaystyle\sum}\limits_{k=1}^{(t-3)/2}a_{2k};$
(4)

(ii) $\alpha^{4}-1$ is a unit  if and only if $t\equiv
1\mathop{\mathrm{m}od}4,$

$a_{t}=-1-2\mathop{\displaystyle\sum}\limits_{k=0}^{(t-5)/4}a_{4k+1},\text{
}a_{t-1}=-1-\mathop{\displaystyle\sum}\limits_{k=1}^{(t-3)/2}a_{2k}\text{ {and
}}a_{t-2}=-\mathop{\displaystyle\sum}\limits_{k=0}^{(t-9)/4}a_{4k+3},$ (5)

or $t\equiv 3\mathop{\mathrm{m}od}4,$

$\text{
}a_{t}=-1-2\mathop{\displaystyle\sum}\limits_{k=0}^{(t-7)/4}a_{4k+3},\text{
}a_{t-1}=-1-\mathop{\displaystyle\sum}\limits_{k=1}^{(t-3)/2}a_{2k}\text{
{and} \ }a_{t-2}=-\mathop{\displaystyle\sum}\limits_{k=0}^{(t-7)/4}a_{4k+1};$
(6)

(iii) $\alpha^{3}-1$ is a unit  if and only if

$t\equiv 0\mathop{\mathrm{m}od}3,\text{ \ }\
a_{t}=-3-2\mathop{\displaystyle\sum}\limits_{k=1}^{(t-3)/3}a_{3k}\ \
\text{{and}\ }\ \
a_{t-1}=-\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\ k\neq
0\mathop{\mathrm{m}od}3\end{subarray}}^{t-2}a_{k},$ (7)

or

$t\equiv 1\mathop{\mathrm{m}od}3,\text{ \ \
}a_{t}=-1-2\mathop{\displaystyle\sum}\limits_{k=0}^{(t-4)/3}a_{3k+1}\text{ \ \
{and} \ \ \ \
}a_{t-1}=-1-\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\
\text{ }k\neq 1\mathop{\mathrm{m}od}3\end{subarray}}^{t-2}a_{k},$ (8)

or

$t\equiv 2\mathop{\mathrm{m}od}3,\
a_{t}=-1-2\mathop{\displaystyle\sum}\limits_{k=0}^{(t-5)/3}a_{3k+2}\ \ \
\text{{and\ }\ }\
a_{t-1}=-1-\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\ \text{
}k\neq 2\mathop{\mathrm{m}od}3\end{subarray}}^{t-2}a_{k}.$ (9)

Proof. (i) Since
$\mathop{\mathrm{N}orm}(\alpha^{2}-1)=\mathop{\mathrm{N}orm}(\alpha-1)\mathop{\mathrm{N}orm}(\alpha+1),$
$\mathop{\mathrm{N}orm}(\alpha+1)=S(-1),$ and $\alpha+1$ is a unit if and only
if $S(-1)=1,$ we see that

$\alpha^{2}-1\text{ is a unit}\Leftrightarrow S(1)=-1=-S(-1),$ (10)

and (4) follows immediately from the relation (10) and the fact that
$(S(1)=-S(-1)=-1)\Leftrightarrow(S(1)+S(-1)=0$ and $S(1)-S(-1)=-2).$ Also, as
mentioned in the proof of [6, Proposition 3.3], we easily get from (10) that
$t\equiv 1\mathop{\mathrm{m}od}2,$ whenever $\alpha^{2}-1$ is a unit. Indeed,
if $\alpha^{2}-1$ is a unit, then $\alpha-1$ is a unit and $(3)$ gives that
$a_{t}$ is odd. On the other hand, if $t$ is even then (10) yields
$4+4a_{2}+\cdot\cdot\cdot+4a_{t-2}+2a_{t}=S(1)+S(-1)=0,$ and so $a_{t}$ is
even.

(ii) First notice that
$\mathop{\mathrm{N}orm}(\alpha^{4}-1)=\mathop{\mathrm{N}orm}(\alpha^{2}+1)\mathop{\mathrm{N}orm}(\alpha^{2}-1),$
and
$\mathop{\mathrm{N}orm}(\alpha^{2}+1)=\mathop{\displaystyle\prod}\limits_{k=1}^{t}(\alpha_{k}^{2}+1)(\alpha_{k}^{-2}+1)=\mathop{\displaystyle\prod}\limits_{k=1}^{t}(\alpha_{k}+i)(\alpha_{k}^{-1}+i)(\alpha_{k}-i)(\alpha_{k}^{-1}-i)=S(-i)S(i)=\left|S(i)\right|^{2},$
where $i^{2}=-1.$ Also, a simple calculation gives that
$S(i)=(2(a_{1}-a_{3}+a_{5}-\cdot\cdot\cdot-a_{t-2})+a_{t})i$ (resp.
$S(i)=(2(a_{1}-a_{3}+a_{5}-\cdot\cdot\cdot+a_{t-2})-a_{t})i),$ when $t\equiv
1\mathop{\mathrm{m}od}4$ (resp. when $t\equiv 3\mathop{\mathrm{m}od}4).$

Now, suppose $\alpha^{4}-1$ unit. Then, $\ \alpha^{2}-1$ is a unit and so by
(4) $t$ is odd and $a_{t}=-1-2(a_{1}+a_{3}+\cdot\cdot\cdot+a_{t-2}).$ Assuming
$t\equiv 1\mathop{\mathrm{m}od}4$ (resp. $t\equiv 3\mathop{\mathrm{m}od}4),$
we obtain from the last two equalities

$S(i)=-(1+4(a_{3}+a_{7}+\cdot\cdot\cdot+a_{t-2}))i\ \ \ \ \ \ \text{(resp.
}S(i)=(1+4(a_{1}+a_{5}+\cdot\cdot\cdot+a_{t-2}))i).$

Moreover, as $\alpha^{2}+1$ is also a unit it follows from the above that

$(1+4(a_{3}+a_{7}+\cdot\cdot\cdot+a_{t-2}))^{2}=\left|S(i)\right|^{2}=1\text{\
\ \ (resp.
}(1+4(a_{1}+a_{5}+\cdot\cdot\cdot+a_{t-2}))^{2}=\left|S(i)\right|^{2}=1),$

$a_{3}+a_{7}+\cdot\cdot\cdot+a_{t-2}=0$ (resp.
$a_{1}+a_{5}+\cdot\cdot\cdot+a_{t-2}=0)$ and $S(i)=-i^{t}.$ Therefore,
$a_{t-2}=-a_{3}-a_{7}-\cdot\cdot\cdot-a_{t-6}$ (resp.
$a_{t-2}=-a_{1}-a_{5}-\cdot\cdot\cdot-a_{t-6})$ and this equality together
with the two ones in (4) yield the desired relations in (5) (resp. in (6)).

Conversely, if the three coefficients $a_{t-2},$ $a_{t-1}$ and $a_{t}$ of $S$
satisfy the equalities in (5) (resp. in (6)), then a direct calculation gives
that $S(1)=-S(-1)=-1,$ $S(i)=-i^{t}$ and so by the above $\alpha^{4}-1$ is a
unit.

(iii) We restrict the proof only to the case $t\equiv
0\mathop{\mathrm{m}od}3,$ since similar arguments lead to the other cases. As
above, we have
$\mathop{\mathrm{N}orm}(\alpha^{3}-1)=\mathop{\mathrm{N}orm}(\alpha^{2}+\alpha+1)\mathop{\mathrm{N}orm}(\alpha-1),$
$(\alpha^{2}+\alpha+1)=(\alpha-j)(\alpha-1/j),$ where $j=e^{i2\pi/3},$
$S(j)=2-a_{1}-a_{2}+2a_{3}+\cdot\cdot\cdot-a_{t-2}-a_{t-1}+a_{t},$ and
$\alpha^{2}+\alpha+1$ is a unit if and only if $S(j)\overline{S(j)}=1.$

Suppose $\alpha^{3}-1$ is a unit. Then, $\alpha-1$ is a unit and so, by (3),
$a_{t}=$ $-3-2(a_{1}+\cdot\cdot\cdot+a_{t-1}).$ It follows from the last two
equalities that
$S(j)=-1-3\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\
k\equiv\pm 1\mathop{\mathrm{m}od}3\end{subarray}}^{t-1}a_{k}.$ Furthermore, as
$\alpha^{2}+\alpha+1$ is a unit we have by the above
$(1+3\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\ k\equiv\pm
1\mathop{\mathrm{m}od}3\end{subarray}}^{t-1}a_{k})^{2}=\left|S(j)\right|^{2}=1,$
$\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\ k\equiv\pm
1\mathop{\mathrm{m}od}3\end{subarray}}^{t-1}a_{k}=0$ and $S(j)=-1.$ Hence,
$a_{t-1}=-\mathop{\displaystyle\sum}\limits_{\begin{subarray}{c}k=1\\\
k\equiv\pm 1\mathop{\mathrm{m}od}3\end{subarray}}^{t-2}a_{k}$ and so
$a_{t}=-3-2\mathop{\displaystyle\sum}\limits_{k=1}^{(t-3)/3}a_{3k}.$ Finally,
if the last two equalities in (7) are true, then a direct calculation shows
that $S(1)=S(j)=-1,$ and hence $\alpha^{3}-1$ is a unit.

Some examples. (1) Suppose $\alpha^{2}-1$ is a unit. Then, (4) asserts that
$t\geq 3,$ and the minimal polynomial of $\alpha$ is of the form

$F_{a}(x):=(x^{6}+1)-a(x^{5}+x)-(x^{4}+x^{2})-(1-2a)x^{3},$

for some $a\in\mathbb{Z},$ when $t=3.$ In fact, it was signaled in [6] that
$F_{a}$ is a Salem polynomial for all $a\geq 0;$ hence there are infinitely
many Salem number of degree $6$ such that $\alpha^{2}-1$ is a unit, and this
confirms Theorem 1(ii) for $(n,t)=(2,3).$

(2) Similarly, if $\alpha^{4}-1$ is a unit, then $t$ is an odd integer at
least $3,$ and the relation (6) gives for $t=3$ that
$(a_{1},a_{2},a_{3})=(0,-1,$ $-1),$ leading to the polynomial $F_{0}.$ Hence,
the Salem number $\alpha_{0}=1.401...$ root of $F_{0}$ is the unique Salem
number $\alpha$ of degree $6$ such that $\alpha^{4}-1$ is a unit ($\alpha_{0}$
is the smallest Salem number of degree $6);$ thus Theorem 1(iii) is optimal
for $n=4.$

(3) By the same way we obtain, using (9), that there is a unique quartic Salem
number $\alpha$ such that $\alpha^{3}-1$ is a unit, namely $\alpha=1.422...$
root of $(1+x^{4})-(x+x^{3})-x^{2}$ ($\alpha$ is the smallest Salem number of
degree $4),$ and so Theorem 1(i) is sharp for $n=3.$ Notice also that each
polynomial of the form

$G_{a}(x):=(x^{6}+1)-a(x^{5}+x)+a(x^{4}+x^{2})-3x^{3}\in\mathbb{Z}[x],$

where $a\geq 3,$ is the minimal polynomial of a Salem number $\alpha$ such
$\alpha^{3}-1$ is a unit. Indeed, a short computation gives that the
polynomial $g_{a}(x):=(x-2)(x+1)(x-a+1)-1$ satisfies
$g_{a}(-1)=-1<0<g_{a}(0)=2a-3,$ $g_{a}(2)=-1<0<g_{a}(1)=2a-5,$
$g_{a}(a-1)=-1<0<g_{a}(a)=(a-2)(a+1)-1,$ and $g_{a}$ is a cubic Salem trace
polynomial. Then, from the identity $G_{a}(x)=x^{3}g_{a}(x+1/x)$ and the
relation (7) we get the claimed result, and hence Theorem 1(i) is true for
$n=t=3.$

(4) Consider the family of polynomials

$H_{a}(x):=(x^{10}+1)-a(x^{9}+x)-a(x^{8}+x^{2})-(1-a)(x^{6}+x^{4})-(1-2a)x^{5},$

where $a\in\mathbb{Z}.$ A simple calculation shows that the polynomial
$h_{a}(x):=$ $x(x^{2}-4)(x+1)(x-(a+1))-1$ satisfies
$H_{a}(x)=x^{5}h_{a}(x+1/x),$ and from Theorem 3 with
$(n,t,C_{4}(x),C(x))=(4,5,x,x+1),$ we obtain for large values of $a$ that
$H_{a}$ is a minimal polynomial of a Salem number $\alpha$ such that
$\alpha^{4}-1.$ Incidentally, the coefficients of $H_{a}$ satisfy the
equalities in (9), and so $\alpha^{3}-1$ is also a unit. Hence, there are
infinitely many Salem numbers of degree $10$ such that $\alpha^{n}-1$ is a
unit for all $n\leq 4.$

In terms of the Salem trace polynomial $T$ of $\alpha,$ Proposition 1 may
rewritten as follows.

Proposition 2. We have the following assertions.

$\alpha-1\text{ {is a unit}}\Leftrightarrow\text{{\ }}T(2)=-1,$ (11)

$\alpha^{2}-1\text{ {is a unit}}\Leftrightarrow(t\equiv
1\mathop{\mathrm{m}od}2\text{ {and }}T(-2)=T(2)=-1),$ (12)
$\alpha^{3}-1\text{{\ is a unit}}\Leftrightarrow T(-1)=T(2)=-1,$ (13)

and

$\alpha^{4}-1\text{ {is a unit}}\Leftrightarrow(t\equiv
1\mathop{\mathrm{m}od}2\text{ {and} }T(-2)=T(0)=T(2)=-1).$ (14)

Proof. Using the relation (1), it is easy to see that
$(2)\Leftrightarrow(11),$ $(4)\Leftrightarrow(12),$ ($(7),$ $(8)$ and
$(9)$)$\Leftrightarrow(13),$ ($(5)$ and $(6)$)$\Leftrightarrow(14),$ and so
Proposition 2 follows from Proposition 1.

A conclusion similar to the previous ones is also obtained when $n=6.$

Proposition 3. Let $\alpha\in\mathbb{T}.$ Then,

$\alpha^{6}-1\text{ {is a unit}}\Leftrightarrow(t\equiv
1\mathop{\mathrm{m}od}2\text{ {and} }T(-2)=T(-1)=T(1)=T(2)=-1).$ (15)

Proof. The same arguments as in the proof of Proposition 1 and Proposition 2
lead to the result.

It follows immediately from Proposition 3, when $\alpha^{6}-1$ is a unit, that
$\deg(T)=t\geq 5,$ and so Theorem 1(ii) is also optimal for $n=6.$ Moreover,
if $Q(x)$ and $R(x)$ denote respectively the quotient and the remainder of the
Euclidean division of $T(x)$ by $(x^{2}-1)(x^{2}-4),$ then
$T(x)=(x^{2}-1)(x^{2}-4)Q(x)+R(x),$ $\deg(R)\leq 3,$ and so $R(x)=-1,$ as (15)
yields $R(-2)=R(-1)=R(1)=R(2)=-1,$ i. e., $T(x)=C_{6}(x)(x^{2}-4)Q(x)-1,$
where $(x^{2}-1)=C_{6}(x).$

Identically, we obtain from (11) and (13) (from (12) and (14)) that the Salem
trace polynomial of a Salem number $\alpha$ such that $\alpha^{n}-1$ is a unit
for some $n\in\\{1,3\\}$ (resp. $n\in\\{2,4\\})$ is of the form
$C_{n}(x)(x-2)Q(x)-1$ (resp. $C_{n}(x)(x^{2}-4)Q(x)-1)$ for some monic
polynomial $Q(x)\in\mathbb{Z}[x]$ (resp. $Q(x)\in\mathbb{Z}[x]$ with odd
degree).

Let us now consider the case $n=5.$ Then, Theorem 1(i) asserts that for any
integer $t\geq 4$ there are infinitely many Salem numbers $\alpha$ of degree
$2t$ such that $\alpha^{5}-1$ is a unit. In fact, this result remains true for
$t=3.$

Proposition 4. If $t\leq 3$ is an integer for which there are infinitely many
Salem numbers $\alpha$ of degree $2t$ such that $\alpha^{5}-1$ is a unit, then
$t=3.$

Proof. First notice that if $\alpha$ is a Salem number of degree $2t\leq 6$
and $\alpha^{5}-1$ is a unit, then $t=3.$ Indeed, assume on the contrary that
$t=2,$ and set $T(x):=x^{2}+ax+b\in\mathbb{Z}[x]$ for the trace polynomial of
$\alpha.$ Then, the relation (1) gives $T(2)=-1$ and
$T(2\cos(2\pi/5))T(2\cos(4\pi/5))\in\\{-1,1\\},$ as $\alpha-1$ and
$\alpha^{4}+\alpha^{3}+\alpha^{2}+\alpha+1$ are units. Hence, $b=-5-2a,$
$5a^{2}+20a+11\in\\{-1,1\\},$ and so there is no quartic Salem number $\alpha$
such that $\alpha^{5}-1$ is a unit, since the last relation is not true for
all $a\in\mathbb{Z}.$

Now suppose $\deg(\alpha)=6$ and $\alpha^{5}-1$ unit. Then, writing the trace
polynomial of $\alpha$ in the form $T(x)=(x^{2}+x-1)(x-2)+ax^{2}+bx+c,$ where
$(a,b,c)\in\mathbb{Z}^{3}$ and
$x^{2}+x-1=(x-2\cos(2\pi/5))(x-2\cos(4\pi/5))=C_{5}(x),$ we have as above
$T(2)=-1$ and $T(2\cos(2\pi/5))T(2\cos(4\pi/5))\in\\{-1,1\\}.$ Hence,
$c=-1-2b-4a,$ $5(a^{2}+b^{2}+a+b+3ab)+1\in\\{-1,1\\},$ and so

$a^{2}+b^{2}+a+b+3ab=0,$ (16)

since $5(a^{2}+b^{2}+a+b+3ab)+1\neq-1.$ On the other hand, a direct
computation gives that the resultant of a polynomial of the form
$P_{(a,b)}(x):=(x^{2}+x-1)(x-2)+ax^{2}+bx-(1+2b+4a),$ where the integers $a$
and $b$ satisfy (16), and the polynomial $(x^{2}+x-1)(x-2)$ is equal to $-1.$

Consequently, to show Proposition 4 it is enough to prove that (16) has
infinitely many solutions $(a,b)\in\mathbb{Z}^{2}$ and the corresponding
polynomials $P_{(a,b)}$ are Salem trace polynomials. For this purpose define a
sequence $(a_{n},b_{n})_{n\geq 0}$ as follows: $a_{0}:=0,$

$b_{n}:=\frac{-(1+3a_{n})+\sqrt{5a_{n}^{2}+2a_{n}+1}}{2},\text{ }$ (17)

and

$a_{n+1}:=\frac{-(1+3b_{n})-\sqrt{5b_{n}^{2}+2b_{n}+1}}{2}.$ (18)

Then, $(a_{0},b_{0})=(0,0),$ $(a_{1},b_{1})=(-1,2),$ $(a_{2},b_{2})=(-6,15),$
and (17) implies that $b_{n}$ is a root of $Q(x,a_{n}),$ where
$Q(x,y)=:x^{2}+x(1+3y)+y+y^{2}=x^{2}+y^{2}+x+y+3xy\in\mathbb{Z}[x,y].$

For each $n\in\mathbb{N}$ consider the relation, say $\Re_{n},$

$(a_{k},b_{k})\in\mathbb{Z}^{2},\text{ \ \ }a_{k}<a_{k-1}\text{\ \ \ and \ \
}b_{k-1}<b_{k},\text{\ for all }k\in\\{1,...n\\}\text{ }$

Clearly, $\Re_{1}$ is true. Suppose that $\Re_{n}$ holds for some $n.$ Because
$Q(x,y)=Q(y,x)$ and $Q(b_{n},a_{n})=0$ we see that $Q(a_{n},b_{n})=0,$ $a_{n}$
is a root of $Q(x,b_{n})$ and so
$x-\alpha:=Q(x,b_{n})/(x-a_{n})\in\mathbb{Z}[x].$ On the other hand, (18)
asserts that $a_{n+1}$ is a root $Q(x,b_{n})$ and so
$a_{n+1}\in\\{a_{n},\alpha\\}\subset\mathbb{Z},$ i. e., $Q(a_{n+1},b_{n})=0.$
Further, the inequalities $0\leq b_{n-1}<b_{n}$ yield, by (18), that $a_{n}>$
$a_{n+1}$ and $a_{n+1}=\alpha.$ Identically, the equality $Q(a_{n+1},b_{n})=0$
yields $b_{n}$ is a root of $Q(x,a_{n+1}),$
$x-\beta:=Q(x,a_{n+1})/(x-b_{n})\in\mathbb{Z}[x],$ and
$b_{n+1}\in\\{b_{n},\beta\\}\subset\mathbb{Z},$ as $b_{n+1}$ is a root of
$Q(x,a_{n+1}).$ Also, the inequalities $-1\geq a_{n}>a_{n+1}$ yield, by (17),
that $b_{n}<b_{n+1},$ $b_{n+1}=\beta,$ and hence $\Re_{n+1}$ is true.
Therefore, all pairs $(a_{n},b_{n})$ belong to $\mathbb{Z}^{2},$ are distinct
and satisfy (16).

To conclude it is enough to verify that $P_{(a_{n},b_{n})}$ is a Salem trace
polynomial. Clearly, $P_{(a_{n},b_{n})}(0)=1-2b_{n}-4a_{n}$ and using (17) it
is easy to see that $P_{(a_{n},b_{n})}(0)<0$ when $a_{n}\leq-2;$ thus
$P_{(a_{n},b_{n})}(0)<0$ for all $n\geq 2,$ because we have in this case by
the above $a_{n}\leq a_{2}=-6.$ In a similar way we get
$P_{(a_{n},b_{n})}(1)>0,$ $P_{(a_{n},b_{n})}(2)<0,$ and so the cubic
polynomial $P_{(a_{n},b_{n})}$ has a root in each of the intervals $(0,1),$
$(1,2)$ and $(2,\infty).$ Consequently, $P_{(a_{n},b_{n})}$ is a Salem trace
polynomial for all $n\geq 2,$ since the interval $(0,2)$ can not contain two
quadratic algebraic integers which are conjugates. Finally, a direct
computation shows that the polynomials $P_{(a_{0},b_{0})}$ and
$P_{(a_{1},b_{1})}$ are also Salem trace polynomials.

3\. Proofs of the theorems

Proof of Theorem 1. Let $n$ be a natural number satisfying $n\equiv
1\mathop{\mathrm{m}od}2$ (resp. $n\equiv 2\mathop{\mathrm{m}od}4,$ $n\equiv
0\mathop{\mathrm{m}od}4).$ As mentioned in the introduction the main tool of
this proof is Theorem 3, and so it is enough to find for each integer $d\geq
0$ a monic separable polynomial $D(x)\in\mathbb{Z}[x]$ of degree $d$ (resp. of
degree $2d,$ of degree $2d+1)$ with all roots (if any) in $(-2,2)$ and such
that $\gcd(D(x),C_{n}(x))=1.$ For this purpose we shall mainly use the first
kind Chebyshev polynomials $t_{k},$ defined by the relation

$t_{k}(2\cos\theta)=2\cos k\theta,\text{
}\forall(k,\theta)\in\mathbb{N}\times[0,\pi].$

Clearly, $t_{1}(x)=x,$ $t_{2}(x)=x^{2}-2,$ and
$t_{k+2}(x)=xt_{k+1}(x)-t_{k}(x),$ $\forall k\in\mathbb{N}.$ Also,
$t_{k}(x)\in\mathbb{Z}[x],$ $\deg(t_{k})=k,$ $t_{k}$ is monic, and $t_{k}$ is
separable, since its roots are
$2\cos(\frac{\pi}{2k})>2\cos(\frac{\pi}{2k}+\frac{\pi}{k})>\cdot\cdot\cdot>2\cos(\frac{\pi}{2k}+\frac{\pi(k-1)}{k}).$
Moreover, we have the following.

Lemma 1. If $n\neq 0\mathop{\mathrm{m}od}4,$ then $\gcd(t_{k},C_{n})=1$ for
all $k\in\mathbb{N}.$

Proof. Let $(k,n)\in\mathbb{N}^{2}$ such that $\gcd(t_{k},C_{n})\neq 1.$ Then,
$n\geq 3,$ as $C_{1}(x)=C_{2}(x)=1,$ and there is a root of $C_{n}$ which is
also a root of $t_{k},$ i. e., there is a natural number $l<n/2$ such that
$t_{k}(2\cos(2l\pi/n))=0.$ Hence, $2\cos(2kl\pi/n)=0$ and so
$2kl\pi/n=\pi/2+\pi m$ for some $m\in\mathbb{Z};$ thus $\ 4kl=n(1+2m)$ and
$n\equiv 0\mathop{\mathrm{m}od}4.$

Lemma 2. Let $(n,m)\in\mathbb{N}^{2}.$ Then,
$\gcd(C_{n},C_{m})=1\Leftrightarrow\gcd(n,m)\in\\{1,2\\}.$

Proof. The result follows immediately from the definition of the polynomials
$C_{n}$ and the well known identity $\gcd(x^{n}-1,x^{m}-1)=x^{\gcd(n,m)}-1.$

Let us now complete the proof of Theorem 1. If we set $t_{0}(x):=1$ and
$D(x):=t_{d}(x)$ (resp. and $D(x):=t_{2d}(x),$ and $D(x):=C_{4d+3}(x)),$ when
$n\equiv 1\mathop{\mathrm{m}od}2$ (resp. when $n\equiv
2\mathop{\mathrm{m}od}4,$ when $n=2^{s}$ for some integer $s\geq 2),$ then we
easily get by Lemma 1 (resp. by Lemma 1, by Lemma 2) that Theorem 1(i) (resp.
Theorem 1(ii), Theorem 1(iii)) is true.

To show the last assertion in Theorem 1, suppose $n\equiv
4\mathop{\mathrm{m}od}8,$ $n\neq 0\mathop{\mathrm{m}od}3$ and
$D(x):=(x-1)t_{2d}(x).$ Then, $D(x)\in\mathbb{Z}[x],$ and $D$ is monic and
separable, since $C_{6}(x)=(x-1)(x+1)$ and Lemma 1 yields
$\gcd(t_{k},C_{6})=1$ for all $k.$ To conclude notice that $\gcd(C_{n},D)=1,$
i. e., $1\neq 2\cos(2l\pi/n)$ and $t_{2d}(2\cos(2l\pi/n))\neq 0$ for all
$l\in\\{1,...,(n-2)/2\\}.$ Indeed, if $1=2\cos(2l\pi/n)$ for some
$l\in\\{1,...,(n-2)/2\\},$ then $\cos(2l\pi/n)=\cos(\pi/3),$ $2l\pi/n=\pi/3,$
$6l=n$ and so $n\equiv 0\mathop{\mathrm{m}od}3.$ Similarly, if
$t_{2d}(2\cos(2l\pi/n))=0$ and $l\in\\{1,...,(n-2)/2\\},$ then
$\cos(4dl\pi/n)=0$ and $4dl\pi/n=\pi/2+\pi m$ for some $m\in\mathbb{Z};$ thus
$8dl=n(1+2m)$ and so $n\equiv 0\mathop{\mathrm{m}od}8.$

Proof of Theorem 2. Let $n\in\mathbb{N}.$ Without loss of generality we may
restrict the proof to the case $n\equiv 0\mathop{\mathrm{m}od}4,$ since for
the other values of $n$ Theorem 2 is an immediate corollary of Theorem 1.
Then, there is an infinite subset $V$ of $\mathbb{N}$ such that
$\gcd(n,4v+3)=1$ for each $v\in V.$ This assertion follows (for instance) from
Dirichlet’s Theorem, saying that there are infinitely many prime numbers of
the form $4v+3.$ Then, Lemma 2 gives that $\gcd(C_{n},C_{4v+3})=1,$ for each
$v\in V,$ and it follows by Theorem 3 with $D:=C_{4v+3}$ that there are
infinitely many Salem numbers $\alpha$ of degree $2t=2(2v+1+(n+4)/2)=4v+n+6$
such that $\alpha^{n}-1$ is a unit. Setting $I:=\\{2v+3+n/2\mid v\in V\\}$ we
obtain the desired result.

Proof of Theorem 3. Set $P(x):=C_{n}(x)D(x)(x-2)(x-a)$ (resp.
$P(x):=C_{n}(x)D(x)(x^{2}-4)(x-a))$ when $n$ is odd (resp. when $n$ is even)
and

$R_{a}(x)=R(x):=P(x)-1.$ (19)

Clearly, the polynomial product $C_{n}D$ is separable with all zeros in the
interval $(-2,2),$ since so are the coprime polynomials $C_{n}$ and $D.$ Also,
$\deg(D)=t-(n+3)/2\geq 0,$ $\deg(C_{n})=$ $(n-1)/2$ (resp.
$\deg(D)=t-(n+4)/2)\geq 0,$ $\deg(C_{n})=$ $(n-2)/2),$ and $\deg(P)=\deg(R)=$
$t\geq 2$ (resp. $t\geq 3).$

Suppose $a\geq 3.$ Then, the polynomial $P$ is separable and its roots, say
$\beta_{1},\beta_{2},...,$ $\beta_{t},$ may be labelled so that

$-2\leq\beta_{1}<\cdot\cdot\cdot<\beta_{t-1}=2<\beta_{t}=a.\text{ \ \ }$

To make clear the remaining part of the proof consider the following two
cases.

Case $t$ odd. Then $t\geq 3.$ For each $k\in\\{1,...,(t-1)/2\\}$ fix an
element $\gamma_{k}$ of the interval $(\beta_{2k-1},\beta_{2k}).$ Then,
$(\gamma_{k}-\beta_{1})>0,$ $...,$ $(\gamma_{k}-\beta_{2k-1})>0,$
$(\gamma_{k}-\beta_{2k})(\gamma_{k}-\beta_{2k+1})\cdot\cdot\cdot(\gamma_{k}-\beta_{t})>0,$
and so $P(\gamma_{k})>0.$ Moreover, if $a>A_{k},$ where

$A_{k}:=\left|\gamma_{k}\right|+\frac{1}{\left|(\gamma_{k}-\beta_{1})(\gamma_{k}-\beta_{2})\cdot\cdot\cdot(\gamma_{k}-\beta_{t-1})\right|},$
(20)

then $\left|\gamma_{k}-\beta_{t}\right|\geq$
$\beta_{t}-\left|\gamma_{k}\right|=a-\left|\gamma_{k}\right|>1/\left|(\gamma_{k}-\beta_{1})(\gamma_{k}-\beta_{2})\cdot\cdot\cdot(\gamma_{k}-\beta_{t-1})\right|,$
and hence $P(\gamma_{k})>1.$ It follows, by (19), that

$R(\gamma_{k})>0>-1=R(\beta_{2k-1})=R(\beta_{2k})$

and the polynomial $R$ has two roots, say $\eta_{2k-1}$ and $\eta_{2k}$ such
that

$\beta_{2k-1}<\eta_{2k-1}<\gamma_{k}<\eta_{2k}<\beta_{2k}.$

Therefore, for each integer $a>A:=\max\\{A_{1},...,A_{(t-1)/2}\\}$ the
polynomial $R$ has $(t-1)$ distinct roots belonging to the interval $(-2,2)$
and the remaining root, say $\eta_{a,t},$ belongs to the interval
$(a,\infty),$ as the monic polynomial $R$ satisfies $R(a)=-1<0.$ In fact,
since the monic polynomial $C_{n}D$ has no root greater than $2,$ we have
$C_{n}(a+1)D(a+1)\geq 1,$ $R(a+1)>0$ and so $\eta_{a,t}\in(a,a+1).$

To complete the proof of this case, suppose $a>A$ and set
$R_{a}(x)=Q_{a}(x)M_{a}(x),$ where $M_{a}$ is the minimal polynomial of
$\eta_{a,t}.$ Then, $Q_{a}(x)\in\mathbb{Z}[x],$ $\deg(Q_{a})\leq t-2$ (recall
that $\eta_{a,t}\in(a,a+1)$ and so $\deg(M_{a})\geq 2),$ $Q_{a}$ is monic, and
the roots of $Q_{a}$ (when $\deg(Q_{a})\geq 1)$ belong to the interval
$(-2,2).$

Because there is a finite number of polynomials in $\mathbb{Z}[x]$ with given
degree and bounded coefficients, the number, say $q,$ of nonconstant monic
polynomials with degree at most $t-2$ and roots in $(-2,2)$ is finite. Notice
also that if $\eta$ is a common zero of $R_{a}$ and $R_{a^{\prime}},$ for some
$(a,a^{\prime})\in(A,\infty)^{2},$ then
$C_{n}(\eta)D(\eta)(\eta-2)(\eta-a)=1=C_{n}(\eta)D(\eta)(\eta-2)(\eta-a^{\prime})$
(resp.
$C_{n}(\eta)D(\eta)(\eta^{2}-4)(\eta-a)=1=C_{n}(\eta)D(\eta)(\eta^{2}-4)(\eta-a^{\prime})),$
$a=a^{\prime}$ and so $\gcd(R_{a},R_{a^{\prime}})=1$ when $a\neq a^{\prime}.$
Consequently, there is at most $q$ polynomials $R_{a}$ which are reducible,
when $a$ runs through the interval $(A,\infty)$ and so there is a constant $B$
such that if $a>B,$ then $R_{a}$ is irreducible, i. e., $R_{a}$ is a trace
polynomial of a Salem polynomial $S_{\alpha},$ where
$\alpha+1/\alpha=\eta_{a,t}.$

Finally, notice that if $\zeta$ is a zero of $x^{n}-1,$ then
$P(\zeta+1/\zeta)=0,$ $R_{a}(\zeta+1/\zeta)=-1,$ and so, by (1),
$S_{\alpha}(\zeta)=-\zeta^{t}.$ Hence, the absolute value of the resultant
$\mathop{\displaystyle\prod}\limits_{\zeta^{n}=1}S_{\alpha}(\zeta)$ of the
polynomials $x^{n}-1$ and $S_{\alpha}(x)$ is equal to $1,$ and consequently
$\alpha^{n}-1$ is a unit.

Case $t$ even.  Since $\deg(P)=t$ and $P$ has no root less than $-2,$ we have
$P(-2)\geq 0.$ Also, [6, Proposition 3.3] gives that $n$ is odd. Thus,
$P(-2)>0,$ $P(-2)\geq 1,$ $R(-2)\geq 0$ and $R$ has a root $\eta_{1}\in$
$[-2,\beta_{1}),$ as $R(\beta_{1})=-1<0.$

Now, if fix for each $k\in\\{1,...,(t-2)/2\\},$ where $t\geq 4,$ an element
$\gamma_{k}$ of the interval $(\beta_{2k},\beta_{2k+1}),$ then we get as above
$(\gamma_{k}-\beta_{1})>0,$ $...,$ $(\gamma_{k}-\beta_{2k})>0,$
$(\gamma_{k}-\beta_{2k+1})\cdot\cdot\cdot(\gamma_{k}-\beta_{t})>0,$ and hence
$P(\gamma_{k})>0.$ Also, if we choose $a>A_{k},$ where $A_{k}$ is defined by
(20), then we obtain $P(\gamma_{k})>1,$

$R(\gamma_{k})>0>-1=R(\beta_{2k})=R(\beta_{2k+1})$

and so the polynomials $R$ has two roots, say $\eta_{2k}$ and $\eta_{2k+1}$
such that

$\beta_{2k}<\eta_{2k}<\gamma_{k}<\eta_{2k+1}<\beta_{2k+1}.$

From this point we conclude in the same way as in the case where $t$ is odd.

References

[1] C. Batut, D. Bernardi, H. Cohen and M. Olivier, User’s Guide to PARI-GP,
Version 2.5.1 (2012).

[2] M. J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugo, M. Pathiaux-Delefosse
and J. P. Schreiber, Pisot and Salem numbers, Birkhäuser Verlag Basel, 1992.

[3] D. W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977), 315-328.

[4] D. W. Boyd, Pisot and Salem numbers in intervals of the real line, Math.
Comp. 32 (1978), 1244-1260.

[5] J. H. Evertse, On equations in S-units and the Thue-Mahler equation,
Invent. Math. 75 (1984), 561-584.

[6] B. H. Gross and C. T. McMullen, Automorphisms of even unimodular lattices
and unramified Salem numbers, J. Algebra. 257 (2008), 265-290.

[7] J. H. Silverman, Exceptional units and numbers of small Mahler measure,
Exp. Math. 4 (1995), 69-83.

[8] T. Zaïmi, Remarks on certain Salem numbers, Arab J. Math. Sc., 7 (2001),
1- 10.

[9] T. Zaïmi, An arithmetical property of powers of Salem numbers, J. Number
Theory 120 (2006), 179- 191.

[10] T. Zaïmi, Comments on Salem polynomials, Arch. Math. 117 (2021), 41–51.

Department of Mathematics and Statistics, College of Science

Imam Mohammad Ibn Saud Islamic University (IMSIU)

P. O. Box 90950

Riyadh 11623 Saudi Arabia

Email<EMAIL_ADDRESS>


# Automatic Mathematical Information Retrieval to Perform Translations up to
Computer Algebra Systems

André Greiner-Petter  University of Konstanz, Konstanz, Germany
<EMAIL_ADDRESS>
(January 2019)

###### Abstract

In mathematics, LaTeX is the de facto standard to prepare documents, e.g.,
scientific publications. While some formulae are still developed using pen and
paper, more complicated mathematical expressions used more and more often with
computer algebra systems. Mathematical expressions are often manually
transcribed to computer algebra systems. The goal of my doctoral thesis is to
improve the efficiency of this workflow. My envisioned method will
automatically semantically enrich mathematical expressions so that they can be
imported to computer algebra systems and other systems which can take
advantage of the semantics, such as search engines or automatic plagiarism
detection systems. These imports should preserve essential semantic features
of the expression.

##  1 Problem & Motivation

The general problem of enriching mathematical expressions with semantic
information and providing lossless translations to computer algebra systems
can be divided into two parts: the translation process and the semantic
enrichment process, hereafter called semantification. The first part was the
focus of my Master´s thesis (see section 3.1). The focus of my doctoral
research will lie on the second part of the problem. Providing extractions
automatically for semantic information of mathematical expressions is
worthwhile for several different tasks and can be examined from different
areas of interests. For example, semantic knowledge strongly improves
mathematical search engines [19, 21]. Imagine someone has the famous
Pythagorean theorem with different variable names, e.g. $q^{2}+p^{2}=r^{2}$.
Finding a connection between this formula and the Pythagorean theorem is still
a difficult task for nowadays search engines. Another, rather new field of
interest is automatic plagiarism detection in mathematical expressions [17].
Detecting plagiarism in a mathematical formula is highly imprecise, if the
semantics of the formula is unknown. As seen in the example above, finding the
connection between $q^{2}+p^{2}=r^{2}$ and the Pythagorean theorem probably
remains unrecognized, which makes it difficult to see a potential plagiarism
in this formula.

During this extended abstract, I will explain the difficulties of an automatic
semantification process on examples related to problems we have for converting
mathematical expressions to computer algebra systems. In the following, I will
give a brief introduction to this workflow and explain its issues in a more
detailed way.

Scientists usually work with word processors to write scientific papers. The
very well-known word processor LaTeX111Since LaTeX is more than just a word
processor, calling LaTeX that way might not be accurate enough. But during our
research, we compared the theoretical concepts behind LaTeX and word
processors such as Microsoft Word and we decided to keep this description. has
become a de facto standard222https://www.latex-project.org/ for this purpose
over the last 30 years [8]. Also, numerous other editors, such as the editor
for Wikipedia articles or Microsoft Word, entirely or partially support LaTeX
expressions. LaTeX, developed by Leslie Lamport, extends the typesetting
system TeX, developed by Donald E. Knuth 1977 [15, p. 559], by a set of
standard macros. Knuth created the TeX system because he was unsatisfied with
the typography of his book, The Art of Computer Programming [15, pp. 5-6, 24].
LaTeX and TeX provide a syntax for printing mathematical formulae in a way a
person would write it by hand. During my research, I will primarily focus on
LaTeX formats.

Systems |  Representations
---|---
Rendered Version | $P_{n}^{(\alpha,\beta)}(\cos(a\Theta))$
Generic LaTeX | `P_n^{(\alpha, \beta)}(\cos(a\Theta))`
Semantic LaTeX | `\JacobiP{\alpha}{\beta}{n}@{\cos@{a\Theta}}`
CAS Maple | `JacobiP(n, alpha, beta, cos(a Theta))`
CAS Mathematica | `JacobiP[n,\[Alpha],\[Beta],Cos[a\[CapitalTheta]]]`
Table 1: Example of different representations for a Jacobi polynomial

Besides that, scientists work with formulae in their papers while they
evaluate special values, create diagrams and find or calculate practical
solutions. Computer algebra systems (CAS) are software tools that allow such
computations on mathematical expressions. Since CAS should be easy to use,
they created their representations with the intent of displaying the results
as intuitively as possible. That approach for representing formulae is not
standardized. That is why most CAS have created their representation, all of
which are different.

Mathematical expressions written in LaTeX, which use only standard libraries
of macros (hereafter called generic LaTeX), do not provide sufficient semantic
information about the used symbols. In contrast to that, a certain level of
semantic information is necessary for CAS to offer correct computations on
input expressions. The exact semantics concluded by readers is based on their
knowledge and in which context the expression is used. Without a translation
tool, a typical workflow for scientists contains different representations for
the same mathematical expression: a representation in a word processor such as
LaTeX, and another representation in a CAS such as Maple or Mathematica. Table
1 in section 2 describes different representations for the same mathematical
expression.

The semantic enrichment problem can be made clear with the description of a
single symbol expression. Consider the Euler-Mascheroni constant represented
by the Greek letter $\gamma$. Without any further information, $\gamma$ is
just a Greek letter, often used to represent this mathematical constant, but
can also be used to represent curve parametrization, and many other things,
such as a variable. In LaTeX, $\gamma$ is represented as `\gamma`. The
equivalent representation in a CAS, such as Maple or Mathematica, depends on
the meaning of $\gamma$. For example, in Maple and Mathematica, the Euler-
Mascheroni constant is represented as `gamma` and `EulerGamma` respectively.
Another more complicated example might be Euler´s number e, which is well-
known to be the constant used for the basis of the natural logarithm.
Translated to Mathematica, it is just a capital `E`, while Maple has no
specific symbol set aside to represent the constant, and one has to evaluate
the exponential function using `exp(1)`.

The above examples indicate two potential problems. A translation with
semantic information might be difficult because one needs to be aware of the
details of the CAS. For instance, a scientist who usually works with
Mathematica might not know that `gamma` in Maple is not only a Greek letter
but also the constant. Therefore, an automated equivalent translation between
representations is desirable. The other problem occurs when the semantic
information is not sufficient or is essentially absent in the expression. For
example, consider the mass-energy equivalence formula $E=mc^{2}$. In plain
LaTeX `E`, `m` and `c` are Latin letters and one has to analyze the context of
each symbol to pick the correct semantic information. It is essential to
clarify the correct semantics before translating an expression to another
representation. Otherwise, `E` might be mistakenly translated as a Latin
letter to ”`E`” in Mathematica instead of `EE` to represent the energy.

##  2 State of the Art

Unfortunately, there is no agreement on how much semantic information is
necessary for an expression to be sufficiently semantic. It always depends on
the task. For example, in most cases, no semantic information is necessary to
render a mathematical expression. For a search engine, providing the name of
the formula might be adequate. But for computations on mathematical
expressions, detailed knowledge about the definitions becomes necessary. In
this context, we suggest that a mathematical expression is sufficiently
semantic when a translation to a CAS becomes feasible.

Before we can start a semantification process, we need a system that is
capable of carrying such information. There are several approaches for
attaching semantic information to symbols or entire expressions. One approach
is the content Mathematical Markup Language (content MathML) [25], which tries
to organize semantic information in an XML document. MathML and content MathML
are widely used for web services because of their simple and easy to parse XML
structure. The JOBAD architecture [10, 6] uses content MathML to create web
documents with access to semantic information. While XML is an appropriate
format for algorithms, it is not convenient for humans.

The National Institute of Standards and Technology (NIST) in Maryland, USA,
has developed a set of LaTeX macros to tie specific character sequences to
well-defined mathematical objects and thereby providing semantic information
within LaTeX expressions. NIST uses these macros for the Digital Library of
Mathematical Functions (DLMF) [18] and the Digital Repository of Mathematical
Formulae (DRMF) [2, 3], an outgrowth of the DLMF project. Thereby we call
these set of macros DLMF/DRMF macros.

Figure 1: Polar plot for the parabolic cylinder function with
$z\left(\phi\right)=2.5e^{i\phi}$ for $\phi\in\left[0,2\pi\right]$. Subfigure
A shows $U\left(0,z\left(\phi\right)\right)$. Subfigure B shows the right-hand
side of DLMF 12.7.10. Subfigure C uses analytic continuation to allow
computations on other branches.

Table 1 gives an overview of several different representations with and
without semantic information for a Jacobi polynomial. The rendered version
illustrates how a scientist would write the expression by hand. As previously
explained, the generic LaTeX expression and the rendered version does not
provide sufficient semantic information in the source, while the DLMF/DRMF
macro, Maple and Mathematica representations are tied to specific definitions.

Besides the noticeable visible differences in used parentheses or the ordering
of the arguments, more complicated and hidden differences may appear for
multi-valued functions. CAS usually define branch cuts to compute principle
values for multi-valued functions. Thereby, CAS implementations of multi-
valued functions are discontinuous. Moreover, the position of these branch
cuts varies from CAS to CAS [5]. It is important that CAS users understand the
position of such cuts [7]. For example, the parabolic cylinder function
$U(a,z)$ is defined as a solution of a second order differential
equation333http://dlmf.nist.gov/12.2#SS1.p1. $U(a,z)$ can also be represented
by other functions, such as the modified Bessel function of the second kind
$K_{\nu}(z)$. One relation between $U$ and $K_{\nu}$ is:

$U(0,z)=\sqrt{\frac{z}{2\pi}}K_{\frac{1}{4}}\left(\frac{1}{4}z^{2}\right),\quad\text{for
}z\in\mathbb{C}.$ (1)

Figure 1 visualizes the problem of this relation using polar plots, for fixed
$r=2.5$ in the polar representation of complex numbers $z=re^{i\phi}$. Figure
1.A draws a curve for $0\leq\phi\leq 2\pi$ of
$U\left(0,z\left(\phi\right)\right)$. The curve is continuous and closed.
Figure 1.B draws the same curve for the principal values of the right-hand
side of the equation as computed by a CAS. The function jumps over branch cuts
at the angles $\phi\in\\{\frac{\pi}{2},\pi,\frac{3\pi}{2}\\}$. Based on the
relation above, Figure 1.B should yield to the same curve as Figure 1.A. The
solution for this problem is analytic continuation which allows computations
on other branches. Figure 1.C shows the right-hand side of the equation with
analytic continuation for $K_{\nu}(z)$. The orange
$\pi\leq\phi\leq\frac{3}{2}\pi$ and blue $\frac{1}{2}\pi\leq\phi\leq\pi$
curves visualize the respective branches, computed with analytic continuation,
while the red $0\leq\phi\leq\frac{1}{2}\pi$, $\frac{3}{2}\pi\leq\phi\leq 2\pi$
curve is the principal value. If CAS users use B rather than C for their
computations, this leads to incorrect conclusions.

Most CAS provide import and export functions to other representations (see
[13, 9, 4]). However, most import functions are only provided for
presentations that already carry semantic information, foremost content
MathML. While import and export functions rarely take care of system-specific
differences [4], such as in case of different branch cut definitions from the
example above, the problems become more serious when the semantic in the
representation is unclear. In this case, the tool has to be able to extract
the semantic information first to allow a translation. To my knowledge, there
is no translation tool available that solves this problem. A typical
workaround is to use translation tools, that can transform mathematical LaTeX
to MathML formats. These tools also contain a lot of problems and cannot
handle the mentioned hidden differences. We already studied those tools and
their accuracy in a paper [22] (see section 3.2).

However, finding ways of an automatic semantification process is an active
topic of research. Nowadays, most semantification algorithms focusing on
natural languages, which leads to the well-studied and broad field of natural
language processing (NLP). In this respect, a relatively new field of interest
is to extend this research for mathematical expressions. Existing approaches
(and work in progress projects) try to adapt the approaches from NLP [23, 24,
16, 19, 26, 3, 20, 1, 21, 22]. Why this adaption process can be error-prone
will be shown by an example I discovered during my research. Our test
formula444https://dlmf.nist.gov/1.5#E2 explains the continuity of a two-valued
function $f$:

$|f(a+\alpha,b+\beta)-f(a,b)|<\epsilon.$ (2)

The surrounding text contains the following part: ”[$\ldots$] for every
arbitrarily small positive constant $\epsilon$ [$\ldots$]”. Basically, every
preceding and following noun of an identifier (in this case $\epsilon$) is a
candidate for a definiens555Latin for a term that describes another term (in
our case symbols or identifiers).. We were using the Stanford
CoreNLP666https://stanfordnlp.github.io/CoreNLP/ to tag words. Unfortunately,
the tagger tagged the word ”constant” as an adjective rather than a noun,
which leads to the wrong assumption that $\epsilon$ has no definiens in the
surrounding text. While this example was an extreme case, the context
analyzation process is usually time-consuming and mostly finds a lot of
possible but not correct pairs of identifiers and definiens, i.e., a high
recall777Fraction of relevant instances that have been retrieved from the
sources over the total amount of relevant instances in the sources. and a low
precision888Fraction of relevant instances among the retrieved instances.
value. Note that a high recall value does not mean there would be no space for
improvements, seen in the $\epsilon$-example above.

##  3 Preliminary Research

My preliminary research is structured in two parts. The research starts with
my Master´s thesis, which I describe in the following subsection 3.1. The
second part focused on analyzing existing tools, creating a comprehensive gold
standard and develope improvement techniques, explained in subsection 3.2.

###  3.1 Translation from semantic LaTeX to CAS

Figure 2: Flow diagram, which explains the translation process between
semantic LaTeX and a CAS. The POM-Tagger parses a LaTeX expression based on
the lexicon files and creates a syntax tree representation. That syntax tree
can be translated node by node to a CAS representation.

I completed my Master´s thesis at the TU Berlin in Germany, NIST, USA, and
Maplesoft, Canada. I focused my research on the translation between special
function and orthogonal polynomial LaTeX expressions with DLMF/DRMF macros and
representations in the Maple and Mathematica CAS. The thesis was also
published in a conference paper [4].

From a scientific point of view, mathematics in LaTeX is primarily a formal
language with an alphabet, words, and rules. LaTeX mathematical expressions
can be described by a context-free grammar, to build a syntactical structure.
We used the Part-of-Math tagger (POM-Tagger) [26], to build a syntax tree of a
mathematical expression written in LaTeX. With a syntax tree, we can translate
an expression, node by node without changing the meaning of the expression.

Figure 2 describes the translation process for the Jacobi polynomial example
in table 1. The POM-Tagger uses the lexicon files to tag each token in the
input expression with additional information. These lexicon files store that
information, that describes the meaning of each symbol. I extended the lexicon
files to provide translation patterns for the semantic DLMF/DRMF macros.
Thereby, the POM-Tagger provides translation patterns for known semantic
macros and also semantic but ambiguous information about other symbols. For
instance, in the lexicon file, the symbol `\beta` indicates that it is a Greek
letter and it is often used to describe the second of three angles in a
triangle, or in case of special functions, it could be the Dirichlet beta
function, and so on and so forth. In our example from table 1, it is just a
variable. Picking the right meaning is essential and one of the goals of my
doctoral thesis.

For the Master´s thesis, we only allowed disambiguated expressions with
DLMF/DRMF macros, to avoid all of the problems previously described. We also
followed the intuitive approach and presumed that the semantic information is
already provided. Consequently, an author has to provide this information
during the writing process. In my Master´s thesis, I attempted to provide
mathematical expression representations with sufficient semantic information
to provide a lossless translation to a CAS. Therefore, we refer to a LaTeX
expression which uses DLMF/DRMF macros as semantic LaTeX.

###  3.2 Extract semantics from generic LaTeX

While the POM-Tagger follows a lexicon-based approach to tag tokens with
semantic information, other approaches already try to analyze the context,
i.e., the surrounding text, of the formula [23, 16, 20, 1]. The Mathematical
Language Processor (MLP) extracts definiens and identifier pairs based on
context analyzation with NLP algorithms.

During my first stay at NII, I researched on existing tools with the
capability of parsing generic LaTeX expressions, such as the POM-Tagger. We
have chosen the content MathML (cMML) as the output format because the
DLMF/DRMF macros are still not publicly available and thereby not supported by
any other tools than the POM-Tagger and LaTeXML, which was originally
developed to create the semantic version of the DLMF [11]. We developed a
comprehensive gold standard of 300 randomly picked formulae from Wikipedia,
the DLMF sources and the NTCIR 12 task [16, 27]. This gold standard contains
manually annotated semantic information in manually corrected cMML files, such
that the cMMLs can be considered as correct and most accurate for the 300
formulae in the gold standard. We compared the MML files generated from
available tools with the cMML files in the gold standard to measure
similarities and accuracy in extracting the identifiers and their meanings in
the formula. Furthermore, we used the MLP to analyze the context of each
formula from the gold standard. As a test state, we implemented some post-
processing algorithms to improve the generated MML files with results from the
MLP analyzation. For example, when the MLP recognizes a symbol as a function
based on the context, we automatically adjust the generated MML based on this
conclusion, which significantly improves the similarity and accuracy [22].

I will continue to research on the MLP pipeline and will combine the results
from the context analyzation approach with the lexicon-based approach from the
POM-Tagger. During our research, we miss a comprehensive API to handle MathML
files effectively. Thereby, we are planning to provide such an API for others
researchers that they do not need to reinvent the wheel and can follow our
already beaten track. The next section will give a brief overview of the
general objectives of my doctoral research.

##  4 Thesis Objectives

###  4.1 Research Goal & Research Objectives

The goal of my doctoral research is to:

Accomplish and evaluate an approach for automatic extraction of sufficient
semantic information for calculations of mathematical expressions in
scientific publications and web pages.

In respect to this, we call semantic information of a mathematical expression
sufficient when a translation to a CAS becomes feasible. To achieve this goal,
I define the following thesis objectives:

  1. 1.

Analyze the strengths and weaknesses of existing semantification approaches of
mathematical expressions.

  2. 2.

Develop a new semantification concept that will improve the current approaches
based on the identified weaknesses.

  3. 3.

Implement the system for an automatic semantification of mathematical
expressions in real-world scientific documents.

  4. 4.

Implement an extension of the system to provide translations to computer
algebra systems.

  5. 5.

Evaluate the developed system by comparing the performance and accuracy, i.e.,
recall and precision values, with existing approaches and take advantage of
evaluation techniques for CAS translations [4].

In the following, I will give a brief overview of the already discovered
problems of existing approaches and explain my new semantification concept.

###  4.2 The Problems

As already mentioned, an automatic semantification process has performance
weaknesses and complicated to realize, because the semantics are rarely
explicitly given in the expression. Consequently, one has to analyze the
context of the expression to find and extract semantic information. While
those extraction approaches usually already have high recall values, they
still can be improved by improving the used NLP techniques, seen in the
$\epsilon$-example from section 2. A possible improvement might take advantage
of dictionary-based approaches. For example, consider a ”constant” as a noun
in mathematical scientific publications. This would lead us to specialized NLP
algorithms for scientific publications. Another problem is, that existing
approaches also ignore the structure of mathematical expressions, which also
contains useful information to conclude semantic information. Integrating
these techniques as a preprocessing step and use specialized NLP algorithms
would improve performances, recall and precision values. The following
subsection will present my new approach that improves existing tools in the
explained way.

###  4.3 Multiple-Scan Approach

Consider a dictionary, that contains information about the meanings of each
symbol and the structure of the formula in respect of this meaning. The
previously described POM-tagger project contains such a dictionary. Assuming
the correct semantic information is given in that dictionary, we need to
eliminate other possible meanings to conclude the correct information. My
approach will take advantage of the existing tools and combine them with the
mentioned dictionary-based approach. The concept will be split into the
following three research objectives:

  1. 1.

narrow down possible meanings only from the expression itself, without
referring to the context of the expression (dictionary-based),

  2. 2.

refine the process with conclusions from the nearby context of the expressions
(dictionary-based preprocessing with context-based refinements), and

  3. 3.

improve the previous process by analyzing not only the nearby context but the
overall topic of the whole scientific paper or book, its references and other
publications by the authors (further expanded context-based refinements).

The behavior of a human reader inspires the idea behind the concept of the
combination of context- and dictionary-based approaches shown in objectives
(1), (2) and (3). The necessary information to get correct and essential
semantics is given somewhere in the context and in the formula itself. If not,
it would be difficult even for a scientific reader to understand the given
expression. The concept is illustrated in figure 3. In the following, I will
explain each objective in a more detailed way.

Figure 3: Conceptual explanation of the multiple scan approach.

Since we will achieve these three objectives by scanning the environment
multiple times, we call this approach the multiple-scan approach. Objective
(1) concentrates on the expression itself, without extracting information from
the context. My proposed approach is to exploit the coherence between the
structure of given formula and its meaning, constructing a Markov logic
network to deduce possible semantic meanings. Therefore, each meaning gets a
probability. If the highest probability is below a given threshold, it would
be necessary to use (2) and (3) for improving these probabilities. Otherwise,
the probability is sufficiently high for concluding semantic information.

Consider the Jacobi polynomial expression in table 1. The given expression has
a superscript, a subscript and the following expression in parentheses. A
leading expression in letters with the following expression in parentheses may
lead us to the conclusion that the leading expression is the name of a
function and the expression in parentheses is its argument. That is also what
Maple does, even when the name of the function is unknown. Additionally, the
first symbol $P$ has a superscript and a subscript. Note that the Meixner-
Pollaczek999Meixner-Pollaczek polynomial: $P_{n}^{(\lambda)}(x;\phi)$,
http://dlmf.nist.gov/18.19#E6 polynomial $P_{n}^{(\lambda)}(x;\phi)$ and the
associated Legendre101010Associated Legendre Function of the First Kind:
$P_{\nu}^{\mu}(x)$, http://dlmf.nist.gov/14.3#E6 function of the first kind
$P_{\nu}^{\mu}(x)$ are also referenced with $P$ and all of these functions has
a superscript, a subscript and an argument. But the Jacobi polynomial assumes
a superscript of two parameters, while the Meixner-Pollaczek polynomial and
the Legendre function just assume one parameter in the superscript.

Objective (2) based on the finding that around 70 percent of the symbolic
elements in scientific papers are denoted in the surrounding text [24]. The
mentioned NLP approaches try to exploit this finding to retrieve the semantic
information for symbols in a formula. In my thesis, I will primarily work with
the MLP approach described in [23, 20]. This approach extracts the symbols
from the formula (called identifiers) and retrieves nouns from the surrounding
text as candidates for definiens of the identifier. The scoring process
assumes that the chance for a correct combination of identifier and definiens
depends on the distance between the identifier and its definiens and the
distance of the identifier to the closest formula that contains this
identifier. I strongly believe we can improve the scoring process with the
conclusions from my first objective above.

If the correct semantic information is still unsure, objective (3) is the last
way to find a solution. Online compendia, such as arXiv, can be used to
discover the overall topic of a scientific paper, the references and the area
of research of the authors. The MCAT search engine developed by Kristianto,
Topic, and Aizawa [16, 19] has performed such and can extract and score
information from the document at the document granularity level. I will try to
add this engine to our software to solve objective (3).

To achieve the objective (1), we will extend the POM-Tagger and its lexicon
files. That is already part of the planned (and currently work in progress)
development of the POM-Tagger. Therefore, I will focus on objectives (2) and
(3), and support the progress of the POM-Tagger collaboratively with the DRMF
project team.

##  5 Planned Research

###  5.1 Short Term Project Proposal

Wikipedia as a highly frequently used lexicon has over 17 million edits every
month111111https://stats.wikimedia.org/v2/#/all-projects (visited Feb. 2018).
During the last two years (since May 2016 until January 2018) 7 million
different formulae have been rendered via Mathoid in Wikipedia, i.e., there
were 7 million edits during the last two years just in mathematical
expressions121212https://github.com/physikerwelt/wikiMath17 (visited Feb.
2018). Wikipedia uses TeX-Markup since 2003 to write and edit mathematical
expressions. Therefore, the Wikipedia word processor is a highly suitable test
environment to add machine learning algorithms.

The planned recommender system will be embedded to the Wikipedia word
processor to learn and train supervised by each editor who modifies or write
mathematical expressions in a Wikipedia article. The planned service would
work as the usual word processor, that gets additional live updated
recommended semantic versions for his mathematical LaTeX input. An editor will
be able to accept, refuse or just ignore the recommended semantic version for
his current input. Based on his acceptance or refusal, the algorithm will
adjust the scores for the recommended presentations.

Since every semantic macro has a generic LaTeX definition, the backward
translation dataset can be used for first training of the algorithm.
Furthermore, the entire DLMF website is implemented in semantic LaTeX and thus
provide an eminently suitable test and training dataset.

###  5.2 Long Term Plans

As mentioned in the previous subsection, I plan to realize a machine learning
algorithm integrated in Wikipedia. I estimate six months (Aug. 2018 - Jan.
2019) for developing the proposed project and additional three months (Feb.
2019 - Apr. 2019) for reviewing and evaluate the results. The results from
this project will then be combined with the results from the previously
developed and optimized MLP pipeline and the results from the first objective,
which will be achieved by the POM-Tagger project.

The following task will be combining the results from the improved MLP project
and the enhanced POM-Tagger project and develop an overall combination
algorithm that takes advantage of both systems. This phase will finalize
objectives (1) and (2) and realize a dictionary-based preprocessing system
with context-based enhancements. I estimate three months for realizing this
task (May 2019 - Jul. 2019).

With the help of the previously acquired findings, I will investigate the
possibility of deducing semantic information from the overall topic of the
context of an input expression. I am planning to use NLP approaches and online
compendia of scientific papers such as arXiv to explore the topic by the
authors and references of the scientific document. Accomplishing objective (3)
will give us the field of a mathematical expression, so that we can fine-tune
our scoring of objectives (1) and (2). I plan to dedicate six months (Aug.
2019 - Jan. 2020) to this task.

Writing the dissertation for a further six months (Jan. 2020 - Jun. 2020) will
complete my doctoral research project.

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