Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula
Authors: Bean Michael A. 
Published: June 2012

In a previous paper, it was shown that if

F
F

is a binary form with complex coefficients having degree

n
⩾
3

n \geqslant 3

and discriminant

D
F

≠
0

{D_F} \ne 0

, and if

A
F

{A_F}

is the area of the region

|

F
(
x
,
y
)

|

⩽
1

\left | {F(x,y)} \right | \leqslant 1

in the real affine plane, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper D Subscript upper F Baseline EndAbsoluteValue Superscript 1 slash n left-parenthesis n minus 1 right-parenthesis Baseline upper A Subscript upper F Baseline less-than-or-slanted-equals 3 upper B left-parenthesis one third comma one third right-parenthesis">

|

D
F

|

1

/

n
(
n
−
1
)

A
F

⩽
3
B
(

1
3

,

1
3

)

{\left | {{D_F}} \right |^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3})

, where

B
(

1
3

,

1
3

)

B(\frac {1} {3},\frac {1} {3})

denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence

{

M
n

}

\{ {M_n}\}

defined by

M
n

=
max

|

D
F

|

1

/

n
(
n
−
1
)

A
F

{M_n} = \max |{D_F}{|^{1/n(n - 1)}}{A_F}

, where the maximum is taken over all forms of degree

n
n

with

D
F

≠
0

{D_F} \ne 0

, is decreasing, and then by showing that

M
3

=
3
B
(

1
3

,

1
3

)

{M_3} = 3B(\frac {1} {3},\frac {1} {3})

. The resulting estimate,

A
F

⩽
3
B
(

1
3

,

1
3

)

{A_F} \leqslant 3B(\frac {1} {3},\frac {1} {3})

for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence

{

M
n

}

\{ {M_n}\}

. By appealing to the theory of hypergeometric functions, it is shown that

M
4

=

2

7

/

6

B
(

1
4

,

1
2

)

{M_4} = {2^{7/6}}B(\frac {1} {4},\frac {1} {2})

and that

M
4

{M_4}

is attained for the form

X
Y
(

X
2

−

Y
2

)

XY({X^2} - {Y^2})

. It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that

A
F

{A_F}

is the perimeter of the polygon corresponding to

F
F

. Based on this correspondence and a representation theorem for

|

D
F

|

1

/

n
(
n
−
1
)

A
F

|{D_F}{|^{1/n(n - 1)}}{A_F}

, it is conjectured that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript n Baseline equals upper D Subscript upper F Sub Subscript n Sub Superscript asterisk Superscript 1 slash n left-parenthesis n minus 1 right-parenthesis Baseline upper A Subscript upper F Sub Subscript n Sub Superscript asterisk">

M
n

=

D

F
n
∗

1

/

n
(
n
−
1
)

A

F
n
∗

{M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}}

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript n Superscript asterisk Baseline left-parenthesis upper X comma upper Y right-parenthesis equals product Underscript k equals 1 Overscript n Endscripts left-parenthesis upper X sine left-parenthesis StartFraction k pi Over n EndFraction right-parenthesis minus upper Y cosine left-parenthesis StartFraction k pi Over n EndFraction right-parenthesis right-parenthesis">

F
n
∗

(
X
,
Y
)
=

∏

k
=
1

n

(
X
sin
⁡

(

k
π

n

)

−
Y
cos
⁡

(

k
π

n

)

)

F_n^*(X,Y) = \prod _{k = 1}^n \left (X \sin \left (\frac {k\pi }{n}\right ) - Y \cos \left (\frac {k\pi }{n}\right )\right )

, and that the limiting value of the sequence

{

M
n

}

\{ {M_n}\}

is

2
π

2\pi

.