A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space
Discrete Comput Geom
A Note on Geometric Embeddings of Simplicial Complexes in a Euclidean Space
I. Novik 0
0 Institute of Mathematics, The Hebrew University , Givat Ram, Jerusalem 91904 , Israel
In this note we prove that if a simplicial complex K can be embedded geometrically in Rm , then a certain linear system of equations associated with K possesses a small integral solution.
In this note we obtain several necessary conditions on a simplicial complex for possessing
a geometric embedding in Rm . We start by briefly describing the motivation for the
Let K be an n-dimensional simplicial complex. It is well known that any such K can
be embedded (even geometrically) in R2nC1. On the other hand not all n-dimensional
complexes are embeddable in R2n. Works of van Kampen [
], Flores [
], Shapiro [
and Wu [
] provide the necessary conditions for an n-dimensional simplicial complex
to possess a piecewise-linear embedding in Rm for n · m · 2n. These conditions
are also sufficient for the case m D 2n, n 6D 2. (For n D 1, m D 2 the sufficiency
follows easily from Kuratowski’s criterion for planarity of graphs. For n D 2, m D 4
Freedman et al. constructed a two-dimensional simplicial complex for which the van
Kampen–Flores conditions hold but which does not admit an embedding into R4 [
famous consequence of these conditions is that the n-dimensional skeleton of a .2n C
2/dimensional simplex cannot be embedded in R2n. For related results see also [
The question we are interested in is when a simplicial complex which is P.L.
embeddable in Rm admits a geometric embedding in Rm .
Definition 1.1. Let f be an embedding of a simplicial complex K in Rm . f is said to
be a geometric embedding iff the image of any simplex ¾ 2 K is a geometric simplex
in Rm .
It is known that embeddability of a complex does not, in general, imply its geometric
embeddability. For example, there is a triangulated Mo¨bius strip constructed by Brehm
] which possesses no geometric embedding in R3 (not even an embedding in which
all edges are straight line segments). For higher-dimensional examples of this kind
]. There is also a three-dimensional manifold constructed by Freedman which
is topologically embeddable in R4, but has no triangulation which is geometrically
embeddable in R4. The most recent result in this area is a closed triangulated orientable
two-dimensional manifold of genus 6 with 12 vertices found by Bokowski and Guedes de
] that has no geometric embedding in R3. Moreover, Bokowski and Guedes
de Oliveira showed that this manifold possesses no geometric embedding in R3 even
after deleting one specific triangle, thus solving the problem, whether for every g there
exists a triangulated closed orientable 2-manifold of genus g which has no geometric
embedding in R3 (see [
] and [
]), for g ¸ 6.
The famous Heawood conjecture settled by Ringel and Youngs [
] asserts that for
any integer n ¸ 7 such that g D .n ¡3/.n ¡4/=12 2 Z there is a triangulation of a closed
orientable two-dimensional manifold of genus g whose 1-skeleton is a complete graph
on n vertices. One of the open conjectures states that if n is sufficiently large, then such
two-dimensional complexes do not admit a geometric embedding in R3. Note that for
these manifolds n D O.g1=2/. On the other hand, the construction due to McMullen et
] shows that for any g there exists a triangulated two-dimensional closed manifold
of genus g on n D O.g=log g/ vertices which is geometrically embeddable in R3.
In this note we associate with every simplicial complex K and integer m a certain
linear system of equations. It follows from the result of van Kampen, Flores, Shapiro, and
Wu that if K is P.L. embeddable in Rm , then this system possesses an integral solution.
We show that if, in addition, K can be embedded geometrically in Rm , then this linear
system possesses a small integral solution. The central idea of the proof is that in the
case of geometric embeddability the intersection numbers involved in the van Kampen–
Flores conditions cannot be arbitrary, but are rather integers with small absolute values. At
present, we have no application of this result (even with some computer experimentation),
but we hope that it can be useful in attacking the above and similar problems.
The rest of the note is organized as follows: in Section 2 we review the necessary
background on obstructions for P.L. embeddability and in Section 3 we state and prove
obstructions for geometric embeddability.
Obstructions for P.L. Embeddability
In this section we review the necessary background on obstructions for P.L.
embeddability. The presentation here relies mostly on Wu’s book [
Let K be an n-dimensional simplicial complex on the vertex set fa1; : : :, aN g. (We
assume that the simplexes of K are oriented by listing the vertices in increasing order).
Two simplexes of K are said to be nondiagonic if they have no vertices in common. The
deleted product of K , K ¤ K , is a subcomplex of K £ K consisting of products of pairs
of nondiagonic simplexes.
Let Cq .K ¤ K / D LfZ.¾1 £ ¾2/: ¾1 £ ¾2 2 K ¤ K ; dim.¾1 £ ¾2/ D qg be a group of
q-dimensional chains of K ¤ K with coefficients in Z. The cells of K ¤ K are assumed
to be oriented as the product of oriented simplexes. Let
be the ordinary boundary operator @: Cq .K ¤ K / ! Cq¡1.K ¤ K /. Let
Cq .K ¤ K / D HomZ.Cq .K ¤ K /; Z/
be the group of q-dimensional cochains of K ¤ K with coefficients in Z and let ± be the
coboundary operator dual to @.
Note that there is an involution t : K ¤ K ! K ¤ K defined by t .¾1 £ ¾2/ D
.¡1/dim ¾1 dim ¾2 .¾2 £¾1/: Using this involution, we can define the group of q-dimensional
antisymmetric cochains of K ¤ K and the group of q-dimensional symmetric cochains:
Caq .K ¤ K / D f¸ 2 Cq .K ¤ K /: t #¸ D ¡¸g;
Csq .K ¤ K / D f¸ 2 Cq .K ¤ K /: t #¸ D ¸g:
Since the ordinary coboundary operator ± commutes with t #, we can define the groups of
cocycles Zaq .K ¤ K / and Zsq .K ¤ K /, groups of coboundaries Baq .K ¤ K / and Bsq .K ¤ K /,
and cohomology groups Haq .K ¤ K / and Hsq .K ¤ K / in the usual way. Given m 2 N
H½qm .K ¤ K / D
½Hsq .K ¤ K /
Haq .K ¤ K /
if m is even,
if m is odd.
In the following we assume that Rm is endowed with a fixed orientation. Let f : K !
Rm be any P.L. map such that f .¾ / \ f .¿ / D ; for any ¾ £ ¿ 2 Skelm¡1.K ¤ K /. Define
a special embedding m-cocycle 'f D 'f .K / as follows:
'f .¾1 £¾2/ D .¡1/dim.¾1/ f .¾1/¢ f .¾2/
for any m-cell ¾1 £¾2 2 K ¤ K ;
where f .¾1/ ¢ f .¾2/ is the index of intersection (or, intersection number) of simplexes ¾1
and ¾2 in Rm . (For the definition and basic properties of the intersection number the reader
is referred to Wu’s book [
], or, for more modern treatment, to [
]. Roughly speaking,
the index of intersection is the algebraic (i.e., including orientation) number of points of
intersection of two singular cells of complementary dimension in a Euclidean space. In
the special case when ¾1 and ¾2 intersect each other transversely and in at most a finite
number of points, we can assign to each intersection point (more precisely, to each point
.x ; y/ 2 ¾1 £ ¾2 such that f .x / D f .y/) a local index of intersection, s.x ; y/ D §1,
which depends on the relative position of f .Vx / and f .Vy / in the oriented Rm , where
Vx and Vy are small oriented neighborhoods of x in ¾1 and y in ¾2, respectively. In this
case the (total) index of intersection is equal to the sum of local indices.)
The following theorem is a version of the van Kampen–Flores theorem (see [
Theorem 2.1. For any simplicial complex K , 'f .K / 2 Z½mm .K ¤ K /, and so 'f .K /
determines a cohomology class, ['f .K /], in H½mm .K ¤ K /. Moreover, if K is embeddable
in Rm , then 'f .K / 2 B½mm .K ¤ K /, that is, ['f .K /] D 0 in H½mm .K ¤ K /.
It turns out that the class ['f .K /] is independent of a P.L. map f . Indeed, orient Rm
and RmC1 by the ordered systems of coordinates .x1; : : : ; xm / and .x1; : : : ; xm ; xmC1/,
respectively. Let g: K ! Rm be another P.L. map such that g.¾ / \ g.¿ / D ; for any
¾ £ ¿ 2 Skelm¡1.K ¤ K /. Define
h: jK j £ I ! Rm £ I
(where I D [0; 1])
h..x ; t // D .t f .x / C .1 ¡ t /g.x /; t /
for x 2 jK j; t 2 I:
By slightly perturbing the vertices of f .K / and g.K /, if necessary, we may suppose that
they are in general position. We can then define the cochain Á D Áh by
Á .¾ £¿ / D h.¾ £ I /¢h.¿ £ I /
for any .m ¡1/-cell ¾ £¿ 2 K ¤ K ;
where h.¾ £ I / ¢ h.¿ £ I / is the index of intersection of cells ¾ £ I and ¿ £ I in RmC1.
(We orient cells f¾ £ I : ¾ 2 K g by orienting I D [0; 1] from 0 to 1.) Then the following
holds (see p. 180 of [
Proposition 2.1. Á 2 C½mm¡1.K ¤ K / and ±Á D 'g ¡ 'f ; and thus the class ['f ] 2 H½mm
is independent of the choice of f .
The following theorem is another (equivalent) version of the van Kampen–Flores theorem
(these versions are equivalent by Proposition 2.1).
Theorem 2.2 [
]. For a simplicial complex K define
8m .K / D ½..11 ¡C tt ##// PP12ff..aaii00 :: :: :: aaiimm00 // ££ ..aajj00 :: :: :: aajjmm00C/1g/g
m D 2m0;
m D 2m0 C 1;
where summations P , P2 are computed over all possible sets of indices .i; j / such that
i0 < j0 < i1 < ¢ ¢ ¢ < im0 < jm0 ; and j0 < i0 < j1 < ¢ ¢ ¢ < im0 < jm0C1, respectively.
Then 8m .K / 2 Z½mm .K ¤ K /. Moreover, if K is embeddable in Rm , then the cohomology
class of 8m .K /, [8m .K /], is equal to 0 in H½mm .K ¤ K /.
Proof. Let a1; : : : ; aN be the vertices of K . Let C .m; N / be an m-dimensional cyclic
polytope with N vertices c1; : : : ; cN 2 Rm (that is, c1 D x .t1/; : : : ; cN D x .tN / are
the points on the moment curve x .t / D .t; t 2; : : : ; t m / and t1 < t2 < ¢ ¢ ¢ < tN ),
and let g: K ! Rm be linear on each simplex of K which maps the vertices of K
to the corresponding vertices of C .m; N /. It is well known (and easy to check) that
'g.K / D §8m .K /, and so the theorem follows from Theorem 2.1.
Obstructions for Geometric Embeddability
In this section we prove that if a simplicial complex K can be embedded geometrically
in Rm , then, in addition to Theorem 2.2, the following holds.
Theorem 3.1. If K can be embedded geometrically in Rm , then there exists a cochain
¸ 2 C½mm¡1.K ¤ K / such that ±.¸/ D 8m and
j¸.¾1 £¾2/j ·
Proof. Let f : K ! Rm be a geometric embedding of K in Rm and let b1; : : : ; bN 2 Rm
be the vertices of f .K /. Let C .m; N / be a cyclic polytope with vertices c1; : : : ; cN 2 Rm ,
and let g: K ! Rm map all vertices of K to the corresponding vertices of C .m; N /
and be linear on all simplexes of K . Since slight perturbations of vertices do not
change the combinatorics of intersections, we can assume without loss of generality
that b1; : : : ; bN; c1; : : : ; cN are generic.
Define h D h. f; g/ and Áh as in Section 2:
h: jK j £ I ! Rm £ I;
h..x ; t // D .t f .x / C .1 ¡ t /g.x /; t / for x 2 jK j; t 2 I I
Áh .¾ 0 £ ¿ 0/ D h.¾ 0 £ I /¢h.¿ 0 £ I /
for ¾ 0 £¿ 0 2 K ¤ K ;
dim.¾ 0 £¿ 0/ D m ¡1:
By Proposition 2.1, Áh 2 C½mm¡1.K ¤ K /. Moreover, since f is an embedding, the special
embedding cocycle of f , 'f , is equal to 0, and so Proposition 2.1 implies that
±.Áh / D 'g ¡ 'f D §8m ¡ 0 D §8m :
if ±.Áh / D 8m ;
to complete the proof, it is sufficient to show that
jh.¾ 0 £ I /¢h.¿ 0 £ I /j ·
for any .m ¡1/-cell ¾ 0 £¿ 0 2 K ¤ K :
The proof of this fact relies on two lemmas. The first lemma estimates the number of
intersection points of h.¾ 0 £ I / with h.¿ 0 £ I /, and the second lemma computes the sign
of the (local) index of intersection at each such point.
Lemma 3.1. For any .m ¡ 1/-cell ¾ 0 £ ¿ 0 2 K ¤ K the number of pairs
..x ; s/; .y; t // 2 .¾ 0 £ I / £ .¿ 0 £ I /;
such that h.x ; s/ D h.y; t / is not greater than m.
Proof. Let e1; e2; : : : ; em be the standard basis of Rm . Given two simplexes
¾ 0 D .ai0 ; aa1 ; : : : ; aik / and ¿ 0 D .aikC1 ; : : : ; aim /
jf..x ; s/; .y; t // 2 .¾ 0 £ I / £ .¿ 0 £ I /: A ± h.x ; s/ D A ± h.y; t /gj · m:
Let f.0; 1/; .d1; 1/; .d2; 1/; : : : ; .dk; 1/g and f.dkC1; 1/; : : : ; .dm; 1/g be the vertices of
A ± h.¾ 0 £ f1g/ and A ± h.¿ 0 £ f1g/, respectively (here di D .di1; : : : ; dim / 2 Rm ).
Since the vertices of A ± h.¾ 0 £ f0g/ and A ± h.¿ 0 £ f0g/ are f.0; 0/, .e1; 0/; : : : ; .ek; 0/g
and f.ekC1; 0/; : : : ; .em; 0/g, respectively, we obtain that if A ± h.x ; s/ D A ± h.y; t /,
then s D t (0 < s < 1) and the barycentric coordinates of x in ¾ 0 and of y in ¿ 0 (we
denote them by ®0; ®1; : : : ; ®k and ¯kC1; : : : ; ¯m , respectively; here k D dim ¾ 0) satisfy
the following equation:
k k m m
s X ®i di C .1 ¡ s/ X ®i ei D s X ¯j dj C .1 ¡ s/ X ¯j ej:
iD1 iD1 jDkC1 jDkC1
Let u D 1 ¡ 1=s. We can rewrite (1) as
X ®i .di ¡ uei/ ¡ X ¯j .dj ¡ uej/ D 0
(where PmjDkC1 ¯j D 1, since ¯kC1; : : : ; ¯m are barycentric coordinates). Hence u is an
eigenvalue of matrix
D D ..d1/>; : : : ; .dm/>/
and .®1; : : : ; ®k ; ¡¯kC1; : : : ; ¡¯m /> is its eigenvector.
Since b1; : : : ; bN; c1; : : : ; cN are generic, all eigenvalues of D are simple, and so for
each real eigenvalue of D, the dimension of the corresponding eigenspace is equal to 1.
Therefore, for each real eigenvalue of D there is at most one eigenvector .®1; : : : ; ®k ;
¡¯kC1; : : : ; ¡¯m /> satisfying P ¯j D 1. Since x 2 ¾ 0 and y 2 ¿ 0 are uniquely
determined by ®1; : : : ; ®k and ¯kC1; : : : ; ¯m , respectively, and s D t is uniquely determined
by u, we obtain that the number of pairs ..x ; s/; .y; t // such that h.x ; s/ D h.x ; t / is
not larger than the total number of eigenvalues of m £ m matrix D, which is equal
1. Let DQ .u/ D D ¡u Im (where Im is the identity matrix and D is the same matrix as in
Lemma 3.1). Let PD.u/ D det DQ .u/ be the value of the characteristic polynomial
of D at u, and let Diu .i D 1; : : : ; m/ be the i th column of DQ .u/.
2. Given matrix B, denote by B¡i the matrix obtained from B by deleting the i th
row, and by B¡i;¡ j the matrix obtained from B by deleting the i th row and the j th
Let ¾ 0, ¿ 0, and A D A¾ 0;¿0 be as in Lemma 3.1 and let u D 1 ¡ 1=s.
Lemma 3.2. For a fixed orientation of RmC1 there exists "Q D ".¾ 0; ¿ 0/ 2 f§1g such
that for each point ..x ; s/; .y; s// 2 .¾ 0 £ I /£.¿ 0 £ I / satisfying A±h.x ; s/ D A±h.y; s/
the index of intersection of ¾ 0 £ I and ¿ 0 £ I at this point has the same sign as "Q ¢ PD0.u/
(where PD0.u/ is the derivative of PD at u D 1 ¡ 1=s).
Proof. If .x ; t / 2 ¾ 0 £ I and .»0; »1; : : : ; »k / are the barycentric coordinates of x in ¾ 0,
then .x ; t / is uniquely determined by »1; : : : ; »k ; t . In these coordinates, the restriction
of the map A ± h to ¾ 0 £ I (we denote this map by U ), is given by
.»1; : : : ; »k ; t / 7 !U
k k !
t X »i di C .1 ¡ t / X »i ei; t :
Similarly, if .y; t / 2 ¿ 0 £ I and .¹kC1; : : : ; ¹m / are the barycentric coordinates of y
in ¿ 0, then .y; t / is uniquely determined by ¹kC1; : : : ; ¹m¡1; t . In these coordinates, the
restriction of the map A ± h to ¿ 0 £ I (we denote this map by V ) is given by
.¹kC1; : : : ; ¹m¡1; t /
7 !V Ãt Ã mX¡1 ¹j .dj ¡ dm/Cdm!C.1 ¡ t /
Ã m¡1 ! !
X ¹j .ej ¡em/Cem ; t :
Since U and V are smooth maps, there exists " 2 f§1g such that for each point
..®1; : : : ; ®k ; s/; .¯kC1; : : : ; ¯m¡1; s// satisfying U .®; s/ D V .¯; s/, the sign of the
index of intersection at this point is equal to the sign of
; : : : ;
; : : : ;
where the first k C 1 derivatives are calculated at .®1; : : : ; ®k ; s/ and the last m ¡ k are
calculated at .¯kC1; : : : ; ¯m¡1; s/.
Let ¯m D 1 ¡ Pm¡1
jDkC1 ¯j and let u D 1 ¡ 1=s. A straightforward calculation shows
µ Diu ¶
Ã k m
X ®i .di ¡ ei/ ¡ X ¯j .dj ¡ ej/; 0
bDy(1) ¡ 1s .®1; : : : ; ®k ; ¡¯kC1; : : : ; ¡¯m ; 0/ :
" ¢ det
X ®i Diu C
X .¡¯i /.Diu ¡ Dmu / D Dm :
Substituting these results in (3), and using the properties of determinant (together with
the fact that s 2 .0; 1/, and so s > 0), we obtain that the sign of the index of intersection
at ..®; s/; .¯; s// is equal to the sign of
; : : : ;
; : : : ;
X.¡1/i ®i det B¡i C
X .¡1/ j .¡¯j / det B¡ j ;
where B is an .m ¡ 1/ £ m matrix .D1u ; : : : ; Dku ; DkuC1 ¡ Dmu ; : : : ; Dmu¡1 ¡ Dmu /.
theSfiancctethUa.t®P; sm/ D V .¯; s/, the numbers u; ®1; : : : ; ®k ; ¯kC1; : : : ; ¯m satisfy (2). Using
jDkC1 ¯j D 1, we can rewrite (2) as
(The penultimate equality follows from the multilinearity of determinant). Substituting
(6)–(8) in (4) we obtain that the sign of the index of intersection at ..®; s/; .¯; s// is
equal to the sign of
.¡1/kCm¡1" ¢ Xm det ³DQ .u/¡i;¡i ´ D .¡1/kCm " ¢ PD0.u/ D "Q ¢ PD0.u/:
Deleting the i th row from this system and solving the remaining system using Cramer’s
rule we conclude that
D .¡1/m¡i¡1 det ³DQ .u/¡i;¡i
®i det B¡i
D det ¡D1u ; : : : ; Diu¡1; Dmu ; DiuC1; : : : ; Dku ; DkuC1 ¡ Dmu ; : : : ; Dmu¡1 ¡ Dmu ¢¡i
for i D 1; : : : ; k;
¡ ¯i det B¡i D .¡1/m¡i¡1 det DQ .u/¡i;¡i
for i D k C 1; : : : ; m ¡ 1;
and (deleting the mth row)
¯m det B¡m
D 1 ¡ X ¯i det B¡m D det B¡m C
D det ¡D1u ; : : : ; Dku ; DkuC1 ¡ Dmu ; : : : ; Dmu¡1 ¡ Dmu ¢¡m
mX¡1 .¡1/m¡i¡1 det ³DQ .u/¡m;¡i
X .¡1/m¡i¡1 det ¡D1u ; : : : ; Dku ; : : : ; Diu¡1; DiuC1; : : : ; Dmu ¢¡m
D det ¡D1u ; : : : ; Dmu¡1¢¡m D det ³DQ .u/¡m;¡m ´ :
To complete the proof of Theorem 3.1 note that for any two consecutive real roots
u0; u00 of the polynomial PD.u/, the numbers PD0.u0/ and PD0.u00/ have opposite signs.
Therefore, it follows from Lemmas 3.1 and 3.2 that among the points f..x ; s/; .y; t // 2
.¾ 0 £ I / £ .¿ 0 £ I /: h.x ; s/ D h.y; t /g there are at most dm=2e points at which the index
of intersection is equal to C1 and at most dm=2e points at which the index of intersection
is equal to ¡1. Thus, the (total) index of intersection
jÁh .¾ 0 £ ¿ 0/j D jh.¾ 0 £ I / ¢ h.¿ 0 £ I /j ·
l m m :
We now present alternative statements of Theorems 2.2 and 3.1. Given a simplicial
complex K on the vertex set a1; : : : ; aN and an integer m, we can associate with K and
m the following linear system of equations: for every .m ¡ 1/-cell ¾ 0 £ ¿ 0 2 K ¤ K there
is a variable x¾ 0;¿0 and an equation
x¾ 0;¿0 D .¡1/.dim ¾ 0C1/.dim ¿0C1/x¿0;¾ 0 ;
and for every m-cell ¾ £ ¿ 2 K ¤ K , ¾ D .ai0 ; : : : ; aik /, ¿ D .aj0 ; : : : ; ajm¡k / with
k < m ¡ k, or k D m ¡ k and .i0; : : : ; ik / <lex . j0; : : : ; jm¡k / there is an equation A¾;¿ :
XlD0 .¡1/l x.ai0 ;:::;abil ;:::;aik /;¿ C .¡1/k XlD0 .¡1/l x¾;.aj0 ;:::;abjl ;:::;ajm¡k /
D <811 iiff mm DD 22kk Ca1nd anid0 < j0j0<<i0¢ ¢ <¢<j1ik<<¢ ¢ j¢k <; ik < jkC1; (10)
In terms of this linear system Theorems 2.2 and 3.1 are equivalent to:
Corollary 3.1. Let K be a simplicial complex.
1. If K is embeddable in Rm , then the linear system associated with K and m has an
2. If K is geometrically embeddable in Rm , then the linear system associated with
K and m has an integral solution with all variables having the absolute value of
less than dm=2e C 1.
Remarks. 1. Note that if we renumber the vertices of a simplicial complex, then in
general we will obtain another system of linear equations. While the existence of an
integral solution for one of these systems does imply the existence of an integral solution
for the other (as follows from Proposition 2.1), we do not know whether the existence of
a small integral solution for one of these systems implies the existence of a small integral
solution for the other as well.
2. In order to prove that a certain complex K is geometrically nonembeddable in
Rm , using the criterion of Corollary 3.1, one has to show that the corresponding linear
system has no solution in f0; §1; : : : ; §dm=2eg. In particular, in the case of m D 3,
one should check whether there are solutions in f0; §1; §2g. Lutz executed such a
computer check for two complexes known to be geometrically nonembeddable in R3:
Brehm’s M o¨bius strip [
] and the triangulated closed two-dimensional manifold of genus
]. Unfortunately, it turned out that in both of these cases the corresponding linear
systems possess a solution in f0; §1; §2g. Thus our criterion fails to prove geometric
nonembeddability in these cases (at least, using one particular numbering of vertices.)
3. It is interesting to note that the same proof as in Theorem 3.1 shows that if K is any
simplicial complex (embeddable or not embeddable in Rm ), then there exists a cochain
¸Q 2 C½mm¡1.K ¤ K / such that ±. ¸Q/ D 28m and
j ¸Q.¾1 £¾2/j ·
for any .m ¡1/-cell
¾1 £¾2 2 K ¤ K :
In other words, if Ax D b is the linear system associated with K and m, then the system
Ax D 2b does possess a small integral solution.
I am deeply grateful to my thesis adviser Prof. Gil Kalai for helpful discussions and
encouragement. Special thanks go to Frank Lutz for conducting computational experiments
pertaining to this paper.
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