Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles

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by  Nealy Bowden

BASES AND STRUCTURE CONSTANTS OF GENERALIZED
SPLINES WITH INTEGER COEFFICIENTS ON CYCLES
NEALY BOWDEN, SARAH HAGEN, MELANIE KING, AND STEPHANIE REINDERS
5
1
0
2

n
a Abstract. An integer generalized spline is a set of vertex labels on an edge-
J labeled graph that satisfy the condition that if two vertices are joined by an edge,

1 thevertexlabelsarecongruentmodulotheedgelabel. Foundationalworkonthese
3
objects comes from Gilbert, Polster, and Tymoczko, who generalize ideas from

geometry/topology(equivariant cohomologyrings) and algebra (algebraic splines)
]
A to developthe notionof generalized splines. Gilbert, Polster,andTymoczkoprove
R that the ring of splines on a graph can be decomposed in terms of splines on
its subgraphs (in particular, on trees and cycles), and then fully analyze splines
.
h
on trees. Following Handschy-Melnick-Reinders and Rose, we analyze splines on
t
a cycles, in our case integer generalized splines.
m The primary goal of this paper is to establish two new bases for the module
[ of integer generalized splines on cycles: the triangulation basis and the King ba-

sis. Unlike bases in previous work, we are able to characterize each basis element
1
completely in terms of the edge labels of the underlying cycle. As an application
v
6 we explicitly construct the multiplication table for the ring of integer generalized
7 splines in terms of the King basis.
1
0
0
2. 1. Introduction
0
5
1 An integer generalized spline is a set of vertex labels on an edge-labeled graph that
:
v satisfy the condition that if two vertices are joined by an edge, the vertex labels are
i congruent modulo the edge label. (See Definition 2.1 for a precise statement.) Figure
X
1 shows examples of splines on a three-cycle.
r
a
Theterm“spline”comesfromthenameofthethinstripsofwoodusedbyengineersto
model larger constructions like ships or cars. Mathematicians later adopted the term
We are extremely grateful to Julianna Tymoczko, Elizabeth Drellich, and Yue Cao for their
insight and contributions to this paper. We would also like to thank Ruth Haas and Joshua
Bowman for valuable discussions on these topics, and Michael DiPasquale for his thorough review
and comments. This work was supported by Smith College and the National Science Foundation
through the Center for Women in Mathematics [DMS-1143716].
1
2 BOWDEN,HAGEN,KING, ANDREINDERS
1 12 15
5 5 5
3 1 3 2 3 0
2 2 2
1 0 0
Figure 1. The edge labels are t2,5,3u and the sets of vertex labels
t1,1,1u, t0,2,12u, and t0,0,15u each form a spline on the cycle.
torefertopiecewisepolynomialsonpolytopeswiththepropertythatthepolynomials
on the faces agree at their shared edges up to a given degree of smoothness. These
mathematical splines are also used for object-modeling purposes, hence the use of
the name.
Billera pioneered the algebraic study of splines, especially looking into questions
regarding thedimension ofthe moduleof splines [2]. Many peoplecontinued Billera’s
work, including among others, Rose [12, 13] and Haas [7] who worked on identifying
dimension and bases for the module of splines.
Splinetheorydevelopedindependently intopologyandgeometry. Goresky, Kottwitz,
and MacPherson [6], Payne [11], and Bahri, Franz, and Ray [1] constructed equivari-
ant cohomology rings using splines, although they did not use that name.
Gilbert, Polster, and Tymoczko generalize the notion of splines that we use here to
what they call generalized splines [4] . These generalized splines are built on the
dual graph of the polytopes found in classical splines. The work of Billera and Rose
shows that the two constructions (on polytopes or their duals) are equivalent in most
cases, including the cases of classical interest [3].
Cycles turn out to be a particularly important family of graphs to study. Indeed
Gilbert, Polster, and Tymoczko show that the ring of generalized splines on a graph
G can be decomposed in terms of splines on certain trees and cycles in G [4]. They
completely describe splines on trees, while leaving open the investigation of splines
on cycles. Similarly, Rose showed that cycles play a key role in the relations defining
modules of splines [13].
Handschy, Melnick, and Reinders begin analysis of integer generalized splines on
cycles [9]. They prove the existence of a certain flow-up basis (see Definition 2.3),
what we call the smallest-value basis, for splines on cycles, and thus prove that such
BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 3
spline modules are free. They define their basis for arbitrary cycles, but only have
formulas for the leading nonzero elements.
Inthispaperwe introducetwo newbases forthemoduleofinteger generalized splines
on cycles: the triangulation basis and the King basis. Each of these bases is fully
expressible in terms of the edge labels of the cycle, and each has its own strengths.
The triangulation basis, so called because it is constructed from triangulated cycles,
is useful because it exists on arbitrary cycles (Theorem 4.2). The advantage of the
King basis lies in the fact that it is relatively simple to calculate, with the entries
almost constant (Definition 5.1). Although the King basis only exists on cycles with
a pair of relatively prime adjacent edge labels, this restriction is not uncommon in
applications. Infactanevengreaterrestrictionthatalledgelabelsberelativelyprime
is commonly used [5, 10]. The results of our work naturally generalize to principle
ideal domains, which include classical univariate splines and Pru¨fer domains; see
forthcoming work [8].
As an application we present the multiplication table of splines on cycles where the
products of splines are expressed in terms of the King basis. Finding multiplica-
tion tables of equivariant cohomology rings in terms of Schubert bases is the central
problem of Schubert calculus. We view this work as a step in that geometric direc-
tion.
The rest of this paper is organized as follows. In Section 2 we summarize the im-
portant definitions and theorems that we use in our work. In Section 3 we provide
a criterion for the existence of flow-up bases. Sections 4 and 5 are dedicated to
proving the existence of the triangulation basis and King basis respectively. In the
final section we give the multiplication table for the King basis and end with an open
question.
2. Preliminaries
2.1. Results from Handschy, Melnick, and Reinders. Handschy, Melnick, and
Reinders proved a number of results about splines on cycles [9]. Many of their
propositionsandtheorems play key rolesinour proofsregarding triangulationsplines
and King splines. We also use their notation, which we describe in this section.
2.1.1. Basic Definitions. The foundational combinatorial object we study is an edge-
labeled graph, defined here:
4 BOWDEN,HAGEN,KING, ANDREINDERS
Definition 2.1 (Edge-Labeled Graphs). Let G be a graph with k edges ordered
e ,e ,...,e and n vertices ordered v ,...,v . Let ℓ be a positive integer label on
1 2 k 1 n i
edge e and let L “ tℓ ,...,ℓ u be the set of edge labels. Then pG,Lq is an edge-
i 1 k
labeled graph.
Withthisnotationforedge-labeledgraphswehavetheformaldefinitionofsplines:
Definition 2.2 (Splines). A spline on the edge-labeled graph pG,Lq is a vertex-
labeling as follows: if two vertices are connected by an edge e then the two vertex
i
labels are equivalent modulo ℓ . We denote a spline G “ pg ,...,g q where g is the
i 1 n i
label on vertex v for 1 ď i ď n.
i
In this paper we assume the labels g P Z.
i
2.1.2. Flow-Up Classes and the Smallest-Value Basis. Flow-up classes are a partic-
ularly nice class of splines on cycles. They arise geometrically ([5], [10], [14]) and are
an analogue of upper triangular matrices.
Definition 2.3 (Flow-Up Classes). Fix a cycle with edge labels pC ,Lq and fix k
n
with 1 ď k ă n. A flow-up class G on pC ,Lq is a spline with k leading zeros.
k n
We say that a basis whose elements are flow-up classes is a flow-up basis. The
simplest flow-up class is the trivial spline; It exists on any edge-labeled cycle.
Proposition 2.4 (Trivial Splines [9, Prop 2.5]). Fix a cycle with edge labels pC ,Lq.
n
The smallest flow-up class on pC ,Lq is G “ p1,...,1q. Moreover, any multiple of
n 0
G is also a spline. We call the multiples of G trivial splines.
0 0
The following theorem establishes that flow-up classes exist on any edge-labeled
cycle.
Theorem 2.5 (Flow-Up Classes on n-cycles [9, Thrm 4.3]). Fix a cycle with edge
labels pC ,Lq. Let n ě 3 and 1 ď k ă n. There exists a flow-up class G on pC ,Lq.
n k n
The next definition introduces smallest flow-up classes.
Definition 2.6 (Smallest Flow-Up Class). Fix a cycle with edge labels pC ,Lq. The
n
smallest flow-up class G “ p0,...,0,g ,...,g q on pC ,Lq is the flow-up class whose
k k`1 n n
nonzero entries are positive and if G1 “ p0,...,0,g1 ,...,g1q is another flow-up class
k k`1 n
with positive entries then g1 ě g for all entries. By convention we consider
i i
G “ p1,...,1q the smallest flow-up class G .
0 0
BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 5
The following theorem gives an explicit formula for the smallest leading element of
flow-up classes.
Theorem 2.7 (Smallest Leading Element of G [9, Thrm 4.5]). Fix a cycle with edge
k
labels pC ,Lq. Fix n ě 3 and k such that 2 ď k ă n. Let G “ p0,...,0,g ...,g q be a
n k´1 k n
flow-up classonpC ,Lq. Theleadingelementg is amultiple of lcmpℓ ,gcdpℓ ,...,ℓ qq
n k k´1 k n
and there is a flow-up class G with g “ lcmpℓ ,gcdpℓ ,...,ℓ qq.
k´1 k k´1 k n
The smallest flow-up classes exist and form a basis for the set of splines given any
edge-labeled cycle.
Theorem 2.8 (Basisforn-Cycles [9, Thrm4.7]). Fix a cycle with edge labels pC ,Lq.
n
The smallest flow-up classes G ,G ,...,G exist on pC ,Lq and form a basis over
0 1 n´1 n
the integers for the Z-module of splines on pC ,Lq.
n
2.2. Useful Computational Tool. For reasons related to finding an explicit basis
for splines on cycles, we want to find a formula for the value of the variable x in the
following pair of congruences:
x ” y mod a
x ” 0 mod b
#
We note the conditions for when such a solution exists and we give an explicit
formulation for x in terms of y, a, and b provided a solution does exist.
Proposition 2.9. Consider the system of congruences
x ” y mod a
x ” 0 mod b.
#
If this system has a solution then one solution is given by the following formula:
‚ If a “ 1 then x “ b is a solution to the system.
gcdpa,bq
‚ If a ‰ 1 then
gcdpa,bq
b b ´1
x “ y
gcdpa,bq gcdpa,bq
ˆ ˙ˆ ˙modp a q
gcdpa,bq
is a solution to the system.
6 BOWDEN,HAGEN,KING, ANDREINDERS
Proof. The Chinese Remainder Theorem tells us that this system of congruences is
satisfied if and only if y ” 0 mod gcdpa,bq. In what follows we will assume that a
solution exists, and thus that y ” 0 mod gcdpa,bq.
Case 1: Let’s deal first with the case where a “ 1. This condition implies
gcdpa,bq
that gcdpa,bq “ a and so b “ an for some n P Z. Because y ” 0 mod gcdpa,bq by
assumption and gcdpa,bq “ a we have y ” 0 mod a. In other words, y “ am for
some m P Z. Then x “ b satisfies the system of congruences because b is congruent
to zero modulo b and b “ an is congruent to y “ am modulo a.
Case 2: Now suppose a ‰ 1. We can rewrite the system of congruences as
gcdpa,bq
x “ y `as
x “ bt
#
Equate both expressions.
bt “ y `as
Recall that y ” 0 mod gcdpa,bq. This allows us to divide both sides by gcdpa,bq and
get an integer as the result.
b y a
t “ ` s
gcdpa,bq gcdpa,bq gcdpa,bq
ˆ ˙ ˆ ˙
Putting this back into modular form we have
b y a
t “ mod .
gcdpa,bq gcdpa,bq gcdpa,bq
ˆ ˙ ˆ ˙
The integers b and a are relatively prime so we can take the inverse
gcdpa,bq gcdpa,bq
of the first mo´dulo th¯e secon´d. ¯
y b ´1 a
t ” mod .
gcdpa,bq gcdpa,bq gcdpa,bq
ˆ ˙ ˆ ˙
Plug this expression for t into the equation x “ bt:
b b ´1
x “ y .
gcdpa,bq gcdpa,bq
ˆ ˙ˆ ˙modp a q
gcdpa,bq
(cid:3)
This value is a solution to the original system of congruences.
Notice that this second case simplifies enormously if gcdpa,bq “ 1. In this situation
x reduces to:
x “ ybrb´1s
moda
BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 7
3. Basis Condition
Let pG,Lq be an arbitrary graph on n vertices with an arbitrary edge-labeling. Con-
sider a set of flow-up classes G ...G on pG,Lq. In this section we give a necessary
0 n´1
and sufficient condition for this set to form a basis for the module of the splines on
pG,Lq. Any set G ,...,G that meets this basis condition is called a flow-up basis.
0 n´1
Such a basis is useful because linear independence is trivially verified.
Let G ...G be a set of flow-up classes and for each i denote
0 n´1
G “ p0,...,0,gpiq ,...,gpiqq.
i i`1 n
The subscript of each gpiq indicates the entry-position of gpiq in the spline G . The
i
superscript piq is to keep track of the fact that we are working with the flow-up class
G . In much of this paper and in previous work the superscript is suppressed when
i
the flow-up class in question is obvious.
Theorem 3.1 (Basis Condition). The following are equivalent:
‚ The set tG ,...,G u forms a flow-up basis.
0 n´1
‚ For each flow-up spline A “ p0,...,0,a ,...,a q the entry a of A is an
i i`1 n i`1 i
integer multiple of the entry gpiq of G .
i`1 i
Proof. Suppose that G ,...,G forms a flow-up basis for the module of splines on
0 n´1
a graph pG,Lq. Suppose that A “ p0,...,0,a ,...,a q is a spline on pG,Lq with
i i`1 n
exactly i leading zeros. We will show that a “ cgpiq for some c P Z.
i`1 i`1
Since G ,...,G form a basis, we can write A as a linear combination of the
0 n´1 i
splines G ,...,G . The fact that A has i leading zeros implies that the coeffi-
0 n´1 i
cients of G ,...,G must be 0. Thus we have A “ c G `...`c G for some
0 i´1 i i i n´1 n´1
c ,...,c P Z. Consider the pi`1qth entry of the splines on the right-hand side of
i n´1
this equation. Note that G is the only element of G ,...,G with a nonzero entry
i i n´1
in this position. Considering the pi`1qth entry on each side of the equation, we have
a “ c gpiq `c 0`...`c 0 “ c gpiq .
i`1 i i`1 i`1 n´1 i i`1
Now we prove the converse. Let A “ pa ,...,a q be an arbitrary spline on pG,Lq.
1 n
We prove by induction that
j´1
A “ A1 ` c G
j k k
k“0
ÿ
8 BOWDEN,HAGEN,KING, ANDREINDERS
for all 1 ď j ď n where A1 is a spline with (at least) j leading zeros.
j
For our base case, note that by hypothesis we have
a ´c gp0q
n 0 n
.
.
A “ ¨ . ˛`c G
0 0
a ´c gp0q
˚ 2 0 2 ‹
˚ 0 ‹
˚ ‹
˝ ‚
sincea “ c gp0q. LettingA1 “ p0,a ´c gp0q,...,a ´c gp0qqgivesA “ A1 ` 0 c G .
1 0 1 1 2 0 2 n 0 n 1 k“0 k k
Thus our claim holds for j “ 1.
ř
Suppose as our induction hypothesis that we have A “ A1 ` i´1 c G for some
i k“0 k k
1 ď i ď n´1. We can write this as
ř
a1
n
.
.
.
¨ ˛
a1 i´1
A “ i`1 ` c G .
˚ 0 ‹ k k
˚ ‹
˚ .. ‹ kÿ“0
˚ . ‹
˚ ‹
˚ 0 ‹
˚ ‹
˝ ‚
By hypothesis we have that a1 “ c gpiq for some c P Z. So we can write
i`1 i i`1 i
a1 ´c gpiq
n i n
.
.
¨ . ˛
a1 ´c gpiq i
˚ i`2 i i`2 ‹
A “ ˚ 0 ‹` ckGk.
˚ ‹
˚ 0 ‹ k“0
˚ ‹ ÿ
˚ .. ‹
˚ . ‹
˚ ‹
˚ 0 ‹
˚ ‹
˝ ‚
LettingA1 “ p0,...,0,0,a1 ´c gpiq ,...,a1 ´c gpiqqgivesusA “ A1 ` i c G .
i`1 i`2 i i`2 n i n i`1 k“0 k k
By induction we have A “ A1 ` j´1 c G for all 1 ď j ď n. In particulařr we have
j k“0 k k
A “ A1 ` n´1c G . But A1 is a spline with n leading zeros. So A1 “ p0,...,0q.
n k“0 k k n ř n
Thus A “ n´1c G . We conclude that every spline can be written as a linear
combinatiořn okf“G0 ,k..k.,G as desired. (cid:3)
0 n´1
ř
BASES AND STRUCTURE CONSTANTS OF GENERALIZED SPLINES 9
One important observation is that the basis condition is only a condition on the first
nonzero entry of each spline in a set of flow-up classes G ,...,G . This gives us
0 n´1
the following useful corollary:
Corollary 3.2. Suppose the set of flow-up classes tG ,...,G u forms a basis for
0 n´1
the module of splines. Suppose tG1,...,G1 u is a set of flow-up classes for which
0 n´1
for each i the first nonzero entry of G1 equals the first nonzero entry of G . Then the
i i
set tG1,...,G1 u also forms a basis for the module of splines.
0 n´1
4. The Triangulation Splines
Triangulationsplinesformanotherbasisofflow-upclassesforcycles. Theyaresimilar
toHandschy, Melnick, andReinders’ smallest-valueflow-upclassesinthattheleading
nonzero elements of both are the same. However we give a formula for every entry
of the triangulation splines, unlike the smallest-value flow-up classes.
Definition 4.1 (Triangulation Splines). Fix an edge-labeled cycle pC ,Lq. For
n
1 ď k ď n´1 the vector H “ p0,...,0,h ,...,h q has entries as follows:
k k`1 n
‚ h “ lcmpℓ ,gcdpℓ ,...,ℓ qq
k`1 k k`1 n
‚ For k `1 ă i ď n if ℓi´1 “ 1 then h “ gcdpℓ ,...,ℓ q.
gcdpℓi´1,...,ℓnq i i n
‚ For k `1 ă i ď n if ℓi´1 ‰ 1 then
gcdpℓi´1,...,ℓnq
gcdpℓ ,...,ℓ q gcdpℓ ,...,ℓ q ´1
i n i n
h “ h
i i´1ˆgcdpℓi´1,...,ℓnq˙ˆgcdpℓi´1,...,ℓnq˙mod ℓi´1
gcdpℓi´1,...,ℓnq
The next theorem establishes that triangulation splines exist on any edge-labeled
cycle.
Theorem 4.2 (ExistenceofTriangulationSplines). Fix an edge-labeledcycle pC ,Lq.
n
For 1 ď k ď n´1 the vector H is a spline on pC ,Lq.
k n
Proof. Start with an edge-labeled cycle pC ,Lq. For 3 ď k ď n ´ 1 add an edge
n
between vertices v and v as shown in Figure 2. Label the edge between v and v
1 k 1 k
with gcdpℓ ,...,ℓ q. We will show the vector H satisfies all of the edge conditions
k n k
represented by this graph, which implies it satisfies the cycle’s edge conditions in
particular.
10 BOWDEN,HAGEN,KING, ANDREINDERS
ℓ ℓ
n´1 n´1
q q
n n
ℓ ℓ
, q , q
ℓ 1´ ℓn ℓ ℓ 1´ ℓn ℓ
n gcdpℓn cdpℓ4,..., ℓnq 3 n gcdpℓn cdpℓ4,..., ℓnq h3 3
gdpℓ3,..., ℓ2 gdpℓ3,..., ℓ2
gc gc
h
2
ℓ 0 ℓ
1 1
(a) Add edges (b) Base case
Figure 2. Triangulated Cycle
Label vertices v ,...,v zero. Label vertex v with
1 k k`1
h “ lcmpℓ ,gcdpℓ ,...,ℓ qq.
k`1 k k`1 n
The integer h satisfies the edge conditions on the downward edges (edges with
k`1
lower-indexed vertices) at vertex v by construction:
k`1
h ” 0 mod ℓ
k`1 k
h ” 0 mod gcdpℓ ,...,ℓ q
# k`1 k`1 n
This is our base case, and we will label vertices from h to h inductively.
k`2 n´1
Our induction hypothesis is that h ,...,h for k ` 1 ď i ď n´ 1 satisfy the edge
k`1 i
conditions for downward edges. Consider the system of congruences at vertex v
i`1
represented by the edges labeled ℓ and gcdpℓ ,...,ℓ q:
i i`1 n
h ” h mod ℓ
i`1 i i
h ” 0 mod gcdpℓ ,...,ℓ q
# i`1 i`1 n
By the Chinese Remainder Theorem a solution h exists if and only if h ” 0 mod
i`1 i
gcdpℓ ,gcdpℓ ,...,ℓ qq. In other words a solution exists if and only if h ” 0 mod
i i`1 n i
gcdpℓ ,...,ℓ q. By our induction hypothesis h satisfies the downward edge conditions
i n i
at vertex v so in particular h ” 0 mod gcdpℓ ,...,ℓ q. Thus a solution h exists.
i i i n i`1