On a Regular Hypersimplex Inscribed into the Multidimensional Cube
It is proved the existence of a regular hypersimplex inscribed into the (4n - 1)-dimensional cube under the vanishing condition of the resultant of some system of 4n - 1 algebraic equations with 4n - 1 unknown quantities.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Цитувати: | On a Regular Hypersimplex Inscribed into the Multidimensional Cube / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 176-185. — Бібліогр.: 6 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859613236927660032 |
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| author | Medianik, A.I. |
| author_facet | Medianik, A.I. |
| citation_txt | On a Regular Hypersimplex Inscribed into the Multidimensional Cube / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 176-185. — Бібліогр.: 6 назв. — англ. |
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| container_title | Журнал математической физики, анализа, геометрии |
| description | It is proved the existence of a regular hypersimplex inscribed into the (4n - 1)-dimensional cube under the vanishing condition of the resultant of some system of 4n - 1 algebraic equations with 4n - 1 unknown quantities.
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Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 2, pp. 176�185
On a Regular Hypersimplex Inscribed
into the Multidimensional Cube
A.I. Medianik
Mathematical Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov, 61103, Ukraine
E-mail:medianik@ilt.kharkov.ua
Received November 28, 2005
It is proved the existence of a regular hypersimplex inscribed into the
(4n� 1)-dimensional cube under the vanishing condition of the resultant of
some system of 4n� 1 algebraic equations with 4n� 1 unknown quantities.
Key words: multidimensional cube, regular simplex, Hadamard's matrix,
circulant matrix, antipodal n-gons, homogeneous system resultant, necessary
and sa�cient conditions.
Mathematics Subject Classi�cation 2000: 52B, 05B20.
1. Introduction
It is well-known that there is no inscribing into the multidimensional cube,
whose dimension is not equal to 4n�1, of a regular simplex of the same dimension
so, that all vertices of the last were vertices of the cube. As to dimension 4n� 1,
H. Coxeter established already in 1933, the equivalence of this problem to the
question of the existence of Hadamard's matrix of order 4n (see [1, p. 319]).
We introduced notions of Hadamard's matrix of half-circulant type [2, p. 459]
and antipodal n-gons inscribed into the regular (2n � 1)-gon [3, p. 48], and
proved that the half-circulant Hadamard matrix of order 4n exists if and only
if there exist antipodal n-gons inscribed into the regular (2n � 1)-gon (see [3,
Th. 4]). The multidimensional problem about existence of a regular hypersimplex,
inscribed into the (4n� 1)-dimensional cube, reduced thereby to a plane problem
on antipodal n-gon, what makes possible to use the methods of algebraic geometry
for its solution. This is considered in the paper.
c
A.I. Medianik, 2006
On a Regular Hypersimplex Inscribed into the Multidimensional Cube
2. De�nitions of Main Notions and its Characteristics
Hadamard's matrix H of order 4n (every its entry equals �1 and rows are
pairwise orthogonal) is said to be half-circulant if it has the following form:
H =
0
BBBB@
1 � � � 1 � � �
... A
... B
1 � � � �1 � � �
... B
... �A
1
CCCCA : (1)
Here A and B are square circulant matrices of order 2n� 1, more precisely, A is
an usual circulant [4, p. 272], which we will call the right circulant, and B is the
left circulant. If a1; a2; : : : ; a2n�1 are entries of the �rst row of a right circulant A,
then entries of its second and next rows are obtained by the cyclic permutation
of previous row to the right: a2n�1; a1; a2; : : : ; a2n�2; a2n�2; a2n�1; a1; : : : ; a2n�3
and so on. The second and next rows of the left circulant B are obtained from
its �rst row b1; b2; : : : ; b2n�1 by the cyclic permutation of previous row to the left,
namely: b2; b3; : : : ; b2n�1; b1; b3; b4; : : : ; b1; b2 and so on.
Let us consider in a complex plane the unit circle with the centre in the
origin. Points zk; k = 0; 1; : : : ; 2n � 2, where z = e
2�i
2n�1 , lie on this circle and
are vertices of the regular (2n � 1)-gon P2n�1. Let Pn and P
0
n be convex n-gons
inscribed into P2n�1 so, that all its vertices are vertices of P2n�1. We say that
convex n-gons Pn and P
0
n, inscribed into the regular (2n � 1)-gon, are antipodal,
if the total number of their diagonals and sides of the same length equals n for
all admissible lengths. For all this, n-gon Pn is represented by the generating
polynomial pn(z) =
P2n�2
k=0 xkz
k, where xk = 1 if the vertex of P 2n�1 with
number k belongs to Pn, and xk = 0 in otherwise. Respectively, n-gon P
0
n
is
represented by a polynomial p
0
n
(z) =
P2n�2
k=0 x
0
k
z
k. Since Pn and P
0
n
are n-gons,
their generating polynomials have exactly n coe�cients xk and x
0
k
equal 1.
The generating polynomial pn(z) has the property (see [3, Lem. 1])
jpnj2 = n+ 2
n�1X
k=1
dk cos
2�k
2n� 1
;
where dk is the number of equal diagonals and sides of n-gon Pn, for which the
vision angle (from the origin) equals 'k = 2�k
2n�1 , k = 1; 2; : : : ; n � 1. There is
similar equality (with replacement dk by d0
k
) for the generating polynomial p0n(z).
Since for antipodal n-gons Pn and P
0
n by de�nition dk + d
0
k
= n, 1 � k � n� 1,
their generating polynomials satisfy relation jpnj2+ jp0nj2 = n by Theorem 3 from
[3].
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 177
A.I. Medianik
As noted in Introduction, the existence of antipodal n-gons is the necessary
and su�cient condition of existence of a half-circulant Hadamard matrix of or-
der 4n. In this connection, there is a natural question about analytical represen-
tation of the antipodal property of n-gons Pn and P
0
n
. To �nd the representation
we assume that
x0 =
1
p
2n� 1
(y0 +
p
2
n�1X
j=1
yj);
xm =
1
p
2n� 1
[y0 +
p
2
n�1X
j=1
(yj cos
2�mj
2n� 1
+ y2n�1�j sin
2�mj
2n� 1
)]; (2)
x2n�1�m =
1
p
2n� 1
[y0 +
p
2
n�1X
j=1
(yj cos
2�mj
2n� 1
� y2n�1�j sin
2�mj
2n� 1
)];
where m = 1; 2; : : : ; n� 1.
Since x0, xm, x2n�1�m equal 0 or 1, parameters y0; y1; : : : ; y2n�2, by which
they are represent, cannot be arbitrary. We obtain, solving linear system (2) with
respect to these parameters,
y0 =
1
p
2n� 1
2n�2X
i=0
xi;
yj =
r
2
2n� 1
[x0 +
n�1X
m=1
(xm + x2n�1�m) cos
2�jm
2n� 1
]; (3)
y2n�1�j =
r
2
2n� 1
n�1X
m=1
(xm � x2n�1�m) sin
2�jm
2n� 1
:
This can be check of the direct substitution into system (2). Let us denote w0,
wm and w2n�1�m the right hand sides of equations of system (2) and consider
following system of quadratic equations:
y0 =
1
p
2n� 1
2n�2X
i=0
w
2
i ;
yj =
r
2
2n� 1
[w2
0 +
n�1X
m=1
(w2
m + w
2
2n�1�m) cos
2�jm
2n� 1
]; (4)
y2n�1�j =
r
2
2n� 1
n�1X
m=1
(w2
m
� w
2
2n�1�m) sin
2�jm
2n� 1
:
178 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On a Regular Hypersimplex Inscribed into the Multidimensional Cube
We �nd, if we solve it with respect to w
2
i
, i = 0; 1; : : : ; 2n � 2 (as the linear
system!): w2
i
= xi = wi, since coe�cients of system (4) coincide with coe�cients
of system (3) and the right hand sides of equations of system (2) are denoted
w0; wm; w2n�1�m. This means that if parameters y0; y1; : : : ; y2n�2 satisfy system
(4), then w
2
i
= wi for all i = 0; 1; 2; : : : ; 2n � 2, i.e. wi, and that is xi, can take
only integer value 0 and 1. It follows from here that system (4) has with respect to
y0; y1; : : : ; y2n�2 22n�1 real-valued solutions, which are represented by form (3),
where each xi takes values 0 or 1 independently from the rest. Thus the following
assertion is valid.
Lemma 1. The coe�cients of the polynomial p(z) =
P2n�2
k=0 xkz
k, which are
represented by equalities (2), take only two values 0 and 1, if and only if their
parameters y0; y1; : : : ; y2n�2 satisfy conditions (4). All solutions of system (4) are
real-valued, and their total number equals 22n�1.
It should be pointed out that among 22n�1 real solutions of system (4) there are
C
n
2n�1 combinations such, that
P2n�2
i=0 xi = n, which corresponds to convex n-gons
inscribed into the regular (2n�1)-gon, with y0 =
np
2n�1
. Next, if y0; y1; : : : ; y2n�2
and y
0
0; y
0
1; : : : ; y
0
2n�2 are two such solutions of system (4), generating convex n-
gons Pn and P
0
n inscribed into the regular (2n � 1)-gon P2n�1, then they are
antipodal if and only if the conditions
y
2
j + y
2
2n�1�j + y
0 2
j + y
0 2
2n�1�j =
2n
2n� 1
; (5)
are valid for all j = 1; 2; : : : ; n� 1 (see [3, Lem. 3]).
Let w = w(y) = w
3
0+
P
n�1
m=1(w
3
m
+w
3
2n�1�m) be a homogeneous polynomial of
third degree with respect to coordinates of vector y, where w0, wm and w2n�1�m
are again the right hand sides of equations (2).
Lemma 2. System (4) is represented in following equivalent form:
y =
1
3
rw; (6)
where rw is a vector with coordinates @w
@yi
, i = 0; 1; 2; : : : ; 2n� 2.
P r o o f. Since
@w
3
i
@y0
=
3w2
ip
2n�1
for all i = 0; 1; 2; : : : ; 2n � 2, then the �rst
equations in (6) has the form:
y0 =
1
p
2n� 1
2n�2X
i=0
w
2
i ;
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 179
A.I. Medianik
which coincides with the �rst equation of system (4).
Since for 0 < j < n
@w
3
0
@yj
= 3
q
2
2n�1w
2
0,
@w
3
m
@yj
= 3
q
2
2n�1w
2
m
cos 2�mj
2n�1 and
@w
3
2n�1�m
@yj
= 3
q
2
2n�1w
2
2n�1�m cos 2�mj
2n�1 , then every equation of the second group
of equation in (6) has a following form:
yj =
r
2
2n� 1
[w2
0 +
n�1X
m=1
(w2
m
+ w
2
2n�1�m) cos
2�mj
2n� 1
];
that coincides with the second equation of system (4).
Besides for 0 < j < n
@w
3
0
@y2n�1�j
= 0,
@w
3
m
@y2n�1�j
= 3
q
2
2n�1w
2
m
sin 2�mj
2n�1
and
@w
3
2n�1�m
@y2n�1�j
= �3
q
2
2n�1w
2
2n�1�m sin 2�mj
2n�1 , then every equation of the third
equation group in (6) has the following form:
y2n�1�j =
r
2
2n� 1
n�1X
m=1
(w2
m
� w
2
2n�1�m) sin
2�mj
2n� 1
;
that coincides with third equation of system (4). This concludes the proof.
Since w is a homogeneous polynomial of third degree by de�nition, then
by Euler's rule
P2n�2
i=0 yi
@w
@yi
= 3w. Therefore, multiplying equations of system
(6) respectively by coordinates y0; y1; : : : ; y2n�2 of vector y and summing theirs
termwise, we obtain w =
P2n�2
i=0 y
2
i
. Since for n-gon Pn inscribed into the regular
(2n � 1)-gon
P2n�2
i=0 xi = n, then it follows from (3) that S =
P2n�2
i=0 y
2
i
= n,
that is, w = n.
Indeed, we obtain, using trigonometrical formulas and so the identity (after
the changing of summing order) 1
2
+
P
n�1
j=1 cos
2�cj
2n�1 � 0, which is valid for all
integer c 6� 0(mod 2n� 1):
S = n
2
2n�1 +
n�1P
j=1
(y2
j
+ y
2
2n�1�j) =
n
2
2n�1 + 2
2n�1 [(n� 1)x20 +
n�1P
j=1
[2x0
n�1P
m=1
(xm
+x2n�1�m) cos
2�jm
2n�1 +
n�1P
m=1
(x2
m
+ x
2
2n�1�m + 2xmx2n�1�m cos 4�jm
2n�1 )
+2
P
m<s
(xmxs + x2n�1�mx2n�1�s) cos
2�j(m�s)
2n�1 + (xmx2n�1�s + x2n�1�mxs)
� cos
2�j(m+s)
2n�1 ]] = n
2
2n�1 + 2
2n�1 [(n� 1)
2n�2P
i=0
x
2
i
� x0
n�1P
m=1
(xm + x2n�1�m)
�
n�1P
m=1
xmx2n�1�m �
P
m<s
(xmxs + xmx2n�1�s + x2n�1�mxs + x2n�1�mx2n�1�s)]
= n
2
2n�1 + 2
2n�1 [n(n� 1)� 1
2
(
2n�2P
i=0
xi)
2 + 1
2
2n�2P
i=0
x
2
i
] = n
2
2n�1 + 2
2n�1 �
n(n�1)
2
= n:
180 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On a Regular Hypersimplex Inscribed into the Multidimensional Cube
The equation w = n determine some hypersurface F in a a�ne space A2n�1.
If we pass to homogeneous coordinates y0; y1; : : : ; y2n�2; y2n�1, then equation
w � ny
3
2n�1 = 0 represents hypersurface of third order in projective space P 2n�1
(w is homogeneous polynomial of third degree by de�nition). It turns out that the
hypersurface F , representing by equation w = n, is a irredusible smooth hypersur-
face both in a�ne space A2n�1 and in projective space P 2n�1 (see [3, Th. 6)].
The above-mentioned results, obtained mostly in paper [3], allowed us to �nd
following necessary and su�cient conditions of the existence of Hadamard's matrix
of half-circulant type (see. Th. 5).
Theorem 1. A half-circulant Hadamard matrix of order 4n exists if and only
if system (6) has two solutions y = fy0; y1; : : : ; y2n�2g and y0 = fy00; y
0
1; : : : ; y
0
2n�2g
such that y0 = y
0
0 = np
2n�1 and so that the rest coordinates of vectors y and y
0
should satisfy antipodal conditions (5).
The above solutions are obviously coordinates of the points of the cubic sur-
faces w = n.
We will mention one more result from algebraic geometry (see [5, p. 174]),
which we need for the proof of our existence theorems for a regular hypersimplex
inscribed into the (4n� 1)-dimensional cube.
Theorem 2. Let
fi(x0; : : : ; xn) = 0 (i = 1; : : : ; r) (7)
be a system of homogeneous equations with undetermined coe�cients and let
�fi(x0; : : : ; xn) = 0 (i = 1; : : : ; r) (8)
be the system of equations, obtained from (7) under some given specialization of its
coe�cients. Then there exists a �nite system of polynomials d1; : : : ; dk, depending
on coe�cients of equations (7) and possessing following characteristics:
(I) for some integer m
dix
m
0 �
rX
j=1
aij(x0; : : : ; xn)fj(x0; : : : ; xn);
*
where coe�cients of polynomials aij(x0; : : : ; xn) belong to the coe�cient ring of
system (7);
(II) necessary and su�cient condition for the existence of solution of system
(8) in some algebraic extension of the coe�cient �eld is the vanishing of polyno-
mials di under a given specialization of coe�cients.
*Sign � means that sum in the right hand side of this equality consists single summand dix
m
0
(after a reduction of similar terms).
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 181
A.I. Medianik
Polynomials d1; d2; : : : ; dk of the theorem, are called the system of resultants
or resultant forms for a system of homogeneous equations with several unknowns.
3. Existence Theorems
Let us introduce by analogy with the polynomial w = w(y) another polynomial
w
0 = w
0 3
0 +
P
n�1
m=1(w
0 3
m
+w
0 3
2n�1�m), whose w
0
i
are given by the right hand sides of
equalities (2), if coordinates of vector y = fy0; y1; : : : ; y2n�2g in them are replaced
by coordinates of vector y0 = fy00; y
0
1; : : : ; y
0
2n�2g. According to Theorem 1 the
existence of a half-circulant Hadamard matrix of order 4n is equivalent to the
solvability of certain equations. The equations can be represented in the form:
8>>>>>>><
>>>>>>>:
Wi =
@w
@yi
� 3yi = 0; i = 0; 1; 2; : : : ; 2n� 2;
W
0
i
= @w
0
@y0
i
� 3y0
i
= 0; i = 0; 1; 2; : : : ; 2n� 2;
W2n�1 = y0 � np
2n�1 = 0; W
0
2n�1 = y
0
0 �
np
2n�1 = 0;
Yj = y
2
j
+ y
2
2n�1�j + y
0 2
j
+ y
0 2
2n�1�j �
2n
2n�1 = 0;
j = 1; 2; : : : ; n� 1:
(9)
Since wi(y) and w
0
i
(y0) are homogeneous polynomial of third degree with re-
spect to its variables, then a homogeneous system, corresponding to (9), has the
form: 8>>>>>>>><
>>>>>>>>:
�Wi =
@w
@yi
� 3yiy2n�1 = 0; i = 0; 1; 2; : : : ; 2n� 2;
�W 0
i
= @w
0
@y0
i
� 3y0
i
y2n�1 = 0; i = 0; 1; 2; : : : ; 2n� 2;
�W2n�1 = y0 �
ny2n�1p
2n�1 = 0; �W 0
2n�1 = y
0
0 �
ny2n�1p
2n�1 = 0;
�Yj = y
2
j
+ y
2
2n�1�j + y
0 2
j
+ y
0 2
2n�1�j �
2ny2
2n�1
2n�1 = 0;
j = 1; 2; : : : ; n� 1:
(10)
System (10) consists homogeneous equations with respect to 4n�1 unknowns
y0; y1; : : : ; y2n�2; y2n�1; y
0
0; : : : ; y
0
2n�2 of degree less than 3. Therefore, one can
obtain every of them from quadratic form (recpectively, linear form) of 4n � 1
variables under some specialization of its undetermined coe�cients. According to
Theorem 2 there exists a �nite system of polynomials d1; d2; : : : ; dk whit respect
to these coe�cients, possessing by characteristics, indicated in the theorem, which
are resultants of system (10).
Theorem 3. Let d1; d2; : : : ; dk be a �nite resultant system of homogeneous
system (10). If every polynomial d1; d2; : : : ; dk vanishes after the substitution of
corresponding coe�cients of system (10), then one can inscribe a regular simplex
of the same dimension into the (4n� 1)-dimensional cube.
182 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On a Regular Hypersimplex Inscribed into the Multidimensional Cube
P r o o f. Since all resultants of system (10) vanish, then it has nontrivial
solution �y0; : : : ; �y2n�1; �y
0
0; : : : ; �y
0
2n�2 in some algebraic extension of its coe�cient
�eld. We shall prove that this solution is real-valued indeed.
Observe �rst of all that if �y2n�1 = 0, then it follows from the �rst equa-
tion of system (10) that @ �w
@yi
= 0, i = 0; 1; : : : ; 2n � 2, where the bar means
that the solution is substituted into a given partial derivative. Multiplying �Wi
by yi and summing obtained equalities termwise, we have by Euler's rule: 3w �
3y2n�1
P2n�2
i=0 y
2
i
= 0 or after a substitution of the solution: 3 �w�3�y2n�1
P2n�2
i=0 �y2
i
= 0. Since �y2n�1 = 0 by assumption, then �w = 0. That is, the point with coor-
dinates �y0; �y1; : : : ; �y2n�2; 0 belongs to hypersurface F of projective space P 2n�1,
representing by equation W = w � ny
3
2n�1 = 0. Since the homogeneous polyno-
mial w does not depend on the variable y2n�1, then both partial derivative @W
@yi
and @W
@y2n�1
vanish in the indicated point, i.e., the point �y0; �y1; : : : ; �y2n�2; 0 is a
singular point of F . This is impossible, since the hypersurface F is irreducible
and smoth in P
2n�1 by the established above.
Consequently, �y2n�1 6= 0. Thus one can assume that in all equations of sys-
tem (10) we have y2n�1 = 1. But system (10) coincides at y2n�1 = 1 with system
(9). Therefore solution �y0; : : : ; �y2n�2; 1; �y
0
0; : : : ; �y
0
2n�2 of system (10) is the solu-
tion of system (9). And since the �rst two groups of equations Wi = 0 and
W
0
i
= 0 of system (9) coincide with system (6) up to notations, then vectors
�y = f�y0; �y1; : : : ; �y2n�2g and �y0 = f�y00; �y
0
1; : : : ; �y
0
2n�2g are solutions of system (6).
By Lemma 2 system (6) coincides with system (4), whose all solutions are real-
valued by Lemma 1, that is, the original solution of system (10) is real-valued
too.
It follows from last equations of system (9) that the coordinates of vectors �y
and �y0 satisfy the conditions �y0 = �y00 = np
2n�1
so and for any j is true: �y2
j
+
�y22n�1�j+�y0 2
j
+�y0 22n�1�j =
2n
2n�1 . Consequently, vectors �y and �y0 represent solutions
of system (6), satisfying all conditions of Theorem 1. Thus, there exists a half-
circulant Hadamard matrix H of order 4n, having form (1). Removing from H
its �rst column (with entries equals 1), we obtain matrix �H, whose rows are the
coordinates of the vertices of a regular hypersimplex in E
4n�1, inscribed into the
hypercube with edge 2, whose centre coincide with the origin (since rows of any
Hadamard's matrix H are pairwise orthogonal, then the vision angle (from the
origin) for each edge of the indicated hypersimplex is the same ' = arccos �1
4n�1 ).
This concludes the proof.
The resultant system of Theorem 3 consists a �nite number of polynomials.
This number can be very large, especially with increase of n. It happens because
the number of equations of system (10) (which equals 5n�1) exceeds signi�cantly
the number of unknown quantities (4n� 1). But, if both quantities are equal to
each other, then the corresponding resultant system consists a single resultant.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 183
A.I. Medianik
More precisely, there exists such resultant form R that another resultant form,
which belongs to the ideal of resultant forms of a given system of homogeneous
equations, is divided by R [5, p. 185]. In connection with this, we modify system
(10) to the following form:
8>>>><
>>>>:
�Wi =
@w
@yi
� 3yiy2n�1 = 0; i = 0; 1; 2; : : : ; 2n� 2;
�W 0
i
= @w
0
@y
0
i
� 3y0
i
y2n�1 = 0; i = 0; 1; 2; : : : ; 2n� 2;
�W 4
2n�1 +
�W 0 4
2n�1 +
n�1P
j=1
�Y 2
j
= 0;
(11)
where it will be necessary to substitute in place of �W2n�1, �W 0
2n�1 and �Yj their
expressions from (10). Then the number of equations of the modi�ed system will
equal 4n� 1, i.e., equate the number of unknowns.
Theorem 4. Let R be resultant of system (11). If R = 0 after the substitution
of coe�cients of system (11), then one can inscribe a regular simplex of the same
dimension into the (4n� 1)-dimensional cube.
P r o o f. It can be proved �rst as above that system (11) has a real-valued
solution �y0; : : : ; �y2n�1; �y
0
0; : : : ; �y
0
2n�2 with �y2n�1 = 1. Then it follows from the
third equation of system (11) that
�W2n�1 = �y0 � np
2n�1 = 0; �W 0
2n�1 = �y00 �
np
2n�1 = 0;
�Yj = �y2
j
+ �y22n�1�j + �y0 2
j
+ �y0 22n�1�j �
2n
2n�1 = 0; j = 1; 2; : : : ; n� 1;
i.e., the given solution of (11) satis�es the last three equations of (10) too.
Thus, the coordinates of vectors �y = f�y0; �y1; : : : ; �y2n�2g and �y0 = f�y00; �y
0
1; : : : ;
�y02n�2g satisfy the equations (6) and all conditions of Theorem 1, whence the
assartion of our theorem follows. This concludes the proof.
If dimension of considered space is very large, the �nding of even one resultant
is a complex technical task. Therefore the following "negative" result may be more
e�ective.
Theorem 5. A half-circulant Hadamard matrix of order 4n does not exist if
and only if there exists polynomials Ai; A
0
i
; A2n�1; A
0
2n�1; Bj, depending on vari-
ables y0; y1; : : : ; y2n�2; y
0
0; : : : ; y
0
2n�2, and such that we have for nonhomogeneous
system (9)
2n�2X
i=0
(AiWi +A
0
i
W
0
i
) +A2n�1W2n�1 +A
0
2n�1W
0
2n�1 +
n�1X
j=1
BjYj � 1: (12)
184 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
On a Regular Hypersimplex Inscribed into the Multidimensional Cube
P r o o f. If relation (12) is true, then, obviously, Wi;W
0
i
;W2n�1;W
0
2n�1; Yj
cannot vanish simultaneously, i.e., system (9) have no solutions. Then any half-
circulant Hadamard matrix of order 4n cannot exist by Theorem 1 too. Con-
versely, if such matrix does not exist, then system (9) has no solutions by Theo-
rem 1. Consequently, according to Theorem 1 from [5, p. 178], there exist poly-
nomials Ai; A
0
i
; A2n�1; A
0
2n�1; Bj of variables y0; y1; : : : ; y2n�2; y
0
0; : : : ; y
0
2n�2 such
that relation (12) is valid for equations of nonhomogeneous system (9). This
concludes the proof.
R e m a r k 1. The conditions of Th. 4 are satis�es, for example, if the
number 2n� 1 is prime one. This follows from [2, Ths. 1 and 2].
R e m a r k 2. The role of the hypersurface of projective space P
2n�1,
represented by equations w = ny
3
2n�1, in proofs of the existence of a regular
hypersimplex inscribed into the (4n� 1)-dimensional cube, is di�erent from that
of our paper [6]. Indeed, in the present paper the homogeneous equivalents of
algebraic equations of Theorem 1, are considered actually in projective space
P
4n�2, while in [6] they are considered on product of two projective spaces P 2n�1
and P
0 2n�1.
References
[1] W. Ball and H. Coxeter, Mathematical Recreations and Essays. Mir, Moscow, 1986.
(Russian)
[2] A.I. Medianik, Regular Simplex Inscribed into a Cube and Hadamard's Matrix of
Half-Circulant Type. � Mat. �z., analiz, geom. 4 (1997), 458�471. (Russian)
[3] A.I. Medianik, Antipodal n-Gons Inscribed into the Regular (2n�1)-Gon and Half-
Circulant Hadamard Matrices of Order 4n. � Mat. �z., analiz, geom. 11 (2004),
45�66. (Russian)
[4] R. Bellman, Introduction to Matrix Analysis. Nauka, Moscow, 1969. (Russian)
[5] W. Hodge and D. Pedoe, Methods of Algebraic Geometry. Izd-vo Inostr. Lit.,
Moscow, 1972. (Russian)
[6] A.I. Medianik, On Existence of a Regular Hypersimplex Inscribed into the (4n�1)-
Dimensional Cube. � J. Math. Phys., Anal., Geom. 1 (2006), 62�72.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 185
|
| id | nasplib_isofts_kiev_ua-123456789-106590 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-11-28T15:42:31Z |
| publishDate | 2006 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Medianik, A.I. 2016-09-30T20:38:30Z 2016-09-30T20:38:30Z 2006 On a Regular Hypersimplex Inscribed into the Multidimensional Cube / A.I. Medianik // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 176-185. — Бібліогр.: 6 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106590 It is proved the existence of a regular hypersimplex inscribed into the (4n - 1)-dimensional cube under the vanishing condition of the resultant of some system of 4n - 1 algebraic equations with 4n - 1 unknown quantities. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии On a Regular Hypersimplex Inscribed into the Multidimensional Cube Article published earlier |
| spellingShingle | On a Regular Hypersimplex Inscribed into the Multidimensional Cube Medianik, A.I. |
| title | On a Regular Hypersimplex Inscribed into the Multidimensional Cube |
| title_full | On a Regular Hypersimplex Inscribed into the Multidimensional Cube |
| title_fullStr | On a Regular Hypersimplex Inscribed into the Multidimensional Cube |
| title_full_unstemmed | On a Regular Hypersimplex Inscribed into the Multidimensional Cube |
| title_short | On a Regular Hypersimplex Inscribed into the Multidimensional Cube |
| title_sort | on a regular hypersimplex inscribed into the multidimensional cube |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106590 |
| work_keys_str_mv | AT medianikai onaregularhypersimplexinscribedintothemultidimensionalcube |