Low Degree Solutions to Linear Equations with K[x]
Coefficients †
M. KALKBRENER‡
M. SWEEDLER‡
L. TAYLOR
§
For given f1 , . . . , fm ∈ K[x] which are relatively prime we present
P degree bounds on the
ai needed to express 1 and other “low degree” polynomials as
ai fi . This paper gives
an improvement on Kakié’s bound (Kakié, 1976).
1. Introduction
If f1 , . . . , fm ∈ K[x], K a field, are relatively prime then 1 can be expressed as 1 =
P
m
i=1 ai fi . In Kakié (1976) the bound
deg(ai ) < max (deg(fj )) + min (deg(fj )) − deg(fi )
1≤j≤m
1≤j≤m
is given. This bound can also be found in Shiffman (1989). In this paper we obtain the
new bound, Gi ,
G1 ≤ max (deg(fj )) − T + 1
1≤j≤m
and
Gi ≤ min (deg(fj )) − T + 1 for i ∈ {2, . . . , m}
1≤j≤m
on the degree of the ai , where f1 is a polynomial of minimal degree among the fi and no
subset of the fi of cardinality T is relatively prime. We give a class of polynomials for
every m in which this bound is attained and prove sharper bounds in a special case.
The question of how to express “1” fits within a natural vector space consideration. For
a natural number D we define K (D) := {f ∈ K[x] | deg(f ) < D}. Integers D1 , . . . , Dm
and D give a degree isomorphism for f1 , . . . , fm if
λ : K (D1 ) ⊕ · · · ⊕ K (Dm ) −→ K (D) ,
(1.1)
Pm
defined by λ(a1 , . . . , am ) = i=1 ai ·fi , is a vector space isomorphism. The general degree
isomorphism problem is to understand the interrelationship between D1 , . . . , Dm and D.
One possible question is: what is the lowest possible D for given f1 , . . . , fm ? Note that
once a degree isomorphism is achieved by a specific D, then for any larger value E there
is also a degree isomorphism. Simply choose i where Di + deg(fi ) is maximal and replace
Di by Di + E − D. The main thrust of this paper is to develop upper and lower bounds
for this lowest possible D.
† This work has been supported by the U.S. Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute
of Cornell University. Contract DAAL03-91-C-0027.
‡ Mathematical Sciences Institute, Cornell University, Ithaca NY 14853.
§ Department of Defense and Mathematical Sciences Institute, Cornell University.
2
M. Kalkbrener, M. Sweedler and L. Taylor
We have two approaches. One uses easy results about modules. The other is even
easier. Using results about modules we obtain an upper bound for this lowest possible D.
It is this result which gives the improvement on Kakié’s bound
P for expressing “1”. It is
worth noting that even for two polynomials one may have 1 =
ai fi with strictly better
bounds on the ai than for the vector space isomorphism. For example, with f1 := 1 − x2
and f2 := x2 the polynomial 1 can be represented in the form a1 f1 + a2 f2 with constant
ai ’s. However, the lowest possible degree D in the degree isomorphism problem is 4 and
D1 = D2 = 2. In fact, for the case of two polynomials f1 and f2 our results (and Kakié’s)
reduce to the classical
with deg(a1 ) < deg(f2 ) and deg(a2 ) < deg(f1 ), all polynomials of degree
less than the degree of f1 f2 have unique representation a1 f1 + a2 f2 .
Our other approach to studying the lowest possible D in a degree isomorphism gives two
lower bounds. The technique uses only the comparing of degrees and the equating of
dimensions. One of the lower bounds obtained in this way is
Pm
i=1 deg(fi )
.
D ≥
m−1
We present examples to illustrate the concepts being discussed and to demonstrate optimality of various bounds.
In this paper we consider the degree isomorphism problem for relatively prime polynomials only. When g, the gcd of f1 , . . . , fm , is not a constant, the right-hand side of
(1.1) must be replaced by K (D) · g. In this case, D1 , . . . , Dm and D give a degree isomorphism for f1 , . . . , fm if and only if D1 , . . . , Dm and D give a degree isomorphism
for f1 /g, . . . , fm /g. Therefore, we can obtain bounds for the general case by computing
bounds for the special case and subtracting the degree of the gcd of f1 , . . . , fm .
2. Upper bound
Let R be a commutative ring with K ⊆ R a field (R is a K-algebra) and I an ideal
of R. Let r ∈ R. Suppose V is a vector subspace of R such that R = V ⊕ (I : r), where
(I : r) := {a ∈ R | a · r ∈ I}. The ideal generated by elements f1 , . . . , fm ∈ R is denoted
by hf1 , . . . , fm i.
Lemma 2.1. The map λ : V ⊕ I −→ R · r + I defined by λ(v, f ) = v · r + f is an
isomorphism of vector spaces.
Proof. The map
γ : R −→
R·r+I
I
defined by γ(a) = a · r has kernel (I : r) by definition of (I : r). This shows that γ carries
V isomorphically to (R · r + I)/I and it immediately follows that λ is an isomorphism.
2
Low Degree Solutions to Linear Equations with K[x] Coefficients
3
Lemma 2.2. Let {f1 , . . . , fm } ⊂ R, I0 any ideal and Ii := I0 + hf1 , . . . , fi i for i ∈
{1, . . . , m}. Suppose that Vi , for i ∈ {1, . . . , m}, is a subspace of R such that R = Vi ⊕
(Ii−1 : fi ).
(a) The map
λ : I0 ⊕ V1 ⊕ · · · ⊕ Vm −→ Im
Pm
defined by λ(f, v1 , . . . , vm ) = f + i=1 vi · fi is a vector space isomorphism.
(b) The map
Im
λ0 : V1 ⊕ · · · ⊕ Vm −→
I0
P
m
defined by λ0 (v1 , . . . , vm ) = i=1 vi · fi is a vector space isomorphism.
(c) Let W be a vector subspace of Im with W ∩ I0 = {0} and Vi · fi ⊆ W for each
i ∈ {1, . . . , m}. Then
λ00 : V1 ⊕ · · · ⊕ Vm −→ W
P
m
defined by λ00 (v1 , . . . , vm ) = i=1 vi · fi is a vector space isomorphism.
Proof. The proof of (a) immediately follows from the previous lemma by induction and
(b) and (c) follow from (a). 2
These results can be generalized to modules over non-commutative rings and abelian
group complements instead of vector space complements.
In this paper we are interested in an application of the above results to the case
R = K[x]. Let f1 , . . . , fm be polynomials in K[x] such that hf1 , . . . , fm i = K[x]. For
every i ∈ {2, . . . , m} let
Gi := deg(gi ),
where gi ∈ K[x] is such that hgi i = (hf1 , . . . , fi−1 i : fi ),
and
G1 := max (Gi + deg(fi )) − deg(f1 ).
2≤i≤m
The following theorem provides a degree isomorphism with D = max2≤i≤m (Gi +deg(fi ))
= G1 + deg(f1 ) and hence provides an upper bound for this lowest possible D.
Theorem 2.3. The map
λ : K (G1 ) ⊕ · · · ⊕ K (Gm ) −→ K (max2≤i≤m (Gi +deg(fi )))
Pm
defined by λ(v1 , . . . , vm ) = i=1 vi · fi is a vector space isomorphism.
Proof. Let g be an arbitrary polynomial in K[x] of degree G1 , I0 := hg · f1 i and
W := K (G1 +deg(f1 )) . With these definitions, this theorem follows from Lemma 2.2(c). 2
We briefly state the form of the above theorem when f1 , . . . , fm are not relatively
prime. Let g be the gcd of f1 , . . . , fm . With the Gi defined as above, the map
λ : K (G1 ) ⊕ · · · ⊕ K (Gm ) −→ K (max2≤i≤m (Gi +deg(fi ))−deg(g)) · g
4
M. Kalkbrener, M. Sweedler and L. Taylor
defined by λ(v1 , . . . , vm ) =
Pm
i=1
vi · fi is a vector space isomorphism.
Our only general upper bound follows directly from Theorem 2.3.
Corollary 2.4. There exist polynomials a1 , . . . , am ∈ K[x] such that
m
X
ai fi = 1
and
deg(ai ) < Gi for every i ∈ {1, . . . , m}.
i=1
In Kakié (1976) the following bound on the degrees of the ai is given:
deg(ai ) < max (deg(fj )) + min (deg(fj )) − deg(fi ).
1≤j≤m
1≤j≤m
(i = 1, . . . , m)
If m = 2, the bound in Corollary 2.4 and Kakié’s bound are equal. We prove in the
following theorem that our bound is better for each of the ai if no subset of cardinality
2 generates K[x].
Theorem 2.5. Let f1 , . . . , fm be ordered in such a way that
deg(f1 ) = min (deg(fj ))
1≤j≤m
and let T be a natural number such that no subset of the fi of cardinality T is relatively
prime. Then
G1 ≤ max (deg(fj )) − T + 1
1≤j≤m
and
Gi ≤ min (deg(fj )) − T + 1 for i ∈ {2, . . . , m}.
1≤j≤m
Proof. It follows from Theorem 2.3, viewing the range of λ as K (G1 +deg(f1 )) , that
m
X
Gi = deg(f1 ).
i=2
By definition of T , at least T of G2 , . . . , Gm must be positive. Therefore,
X
Gi = deg(f1 ) −
Gj ≤ deg(f1 ) − (T − 1).
(i = 2, . . . , m)
j6=1
j6=i
By definition of G1 ,
G1 = max (Gi + deg(fi )) − deg(f1 ) ≤ max (deg(fi )) − T + 1. 2
2≤i≤m
2≤i≤m
3. Lower bounds
Suppose
λ : K (D1 ) ⊕ · · · ⊕ K (Dm ) −→ K (D) ,
Pm
L
defined by λ(a1 , . . . , am ) = i=1 ai ·fi , is a vector space isomorphism. As λ( j6=i K (Dj ) )
lies in the ideal hf1 , . . . , fi−1 , fi+1 , . . . , fm i, it follows that Di must be greater than or
equal to the degree of gcd(f1 , . . . , fi−1 , fi+1 , . . . , fm ). Since for every i ∈ {1, . . . , m}
D ≥ Di + deg(fi ) ≥ deg(gcd(f1 , . . . , fi−1 , fi+1 , . . . , fm )) + deg(fi ),
(3.1)
Low Degree Solutions to Linear Equations with K[x] Coefficients
5
we obtain the lower bound
D ≥
max (Di + deg(fi )) ≥
1≤i≤m
max (deg(gcd(f1 , . . . , fi−1 , fi+1 , . . . , fm )) + deg(fi )).
1≤i≤m
(3.2)
Since
m
X
Di = D
i=1
we can sum the m inequalities D ≥ Di + deg(fi ) in (3.1) and get the bound
Pm
deg(fi )
.
D ≥ i=1
m−1
(3.3)
Example 3.1. Let q1 , q2 , q3 be linear polynomials and d1 , d2 , d3 polynomials of degree
2. We assume that these 6 polynomials are pairwise relatively prime and that d1 , d2 , d3
are K-linearly independent. Let
f1 := q2 · q3 · d1 ,
f2 := q1 · q3 · d2 ,
f3 := q1 · q2 · d3 .
Then
λ : K (2) ⊕ K (2) ⊕ K (2) −→ K (6)
P3
defined by λ(a1 , a2 , a3 ) = i=1 ai · fi is a vector space isomorphism. In this case bound
(3.3) is attained and is strictly greater than bound (3.2).
Example 3.2. Let q1 , q2 , q3 be linear polynomials and d a polynomial of degree 2. We
assume that these 4 polynomials are pairwise relatively prime. Let
f1 := q2 · q3 ,
f2 := q1 · q3 ,
f3 := q1 · q2 · d.
Then, by Theorem 2.3,
λ : K (3) ⊕ K (1) ⊕ K (1) −→ K (5)
P3
defined by λ(a1 , a2 , a3 ) = i=1 ai · fi is a vector space isomorphism. In this case bound
(3.2) is attained and is strictly greater than bound (3.3).
4. A closer analysis
Not surprisingly, the ideals generated by all but one of the fi play a significant role in
a finer analysis. This finer analysis naturally ties in to the method of partial fractions.
For i ∈ {1, . . . , m} define qi := gcd(f1 , . . . , fi−1 , fi+1 , . . . , fm ). Since hf1 , . . . , fm i = K[x],
the polynomials q1 , . . . , qm are pairwise relatively prime. Therefore, q1 · · · qi−1 ·qi+1 · · · qm
divides fi . Let
fi
.
di :=
q1 · · · qi−1 · qi+1 · · · qm
Lemma 4.1. Let i ∈ {2, . . . , m}. The ideal (hf1 , . . . , fi−1 i : fi ) is generated by the polynomial
qi · gcd(d1 , . . . , di−1 )
.
gcd(d1 , . . . , di−1 , di )
6
M. Kalkbrener, M. Sweedler and L. Taylor
Proof. Since hf1 , . . . , fm i = K[x], qj and dj are relatively prime as are fj and qj for
every j ∈ {1, . . . , m}. Therefore, the ideal hf1 , . . . , fi−1 i is generated by
gcd(d1 , . . . , di−1 ) ·
m
Y
qj
j=i
and the ideal (hf1 , . . . , fi−1 i : fi ) is generated by
qi · gcd(d1 , . . . , di−1 )
. 2
gcd(d1 , . . . , di−1 , di )
Let Q :=
Pm
i=1
deg(qi ). With this Q, the bound (3.2) becomes
D ≥ Q + max (deg(di )).
(4.1)
1≤i≤m
The following proposition shows that this lower bound is achieved if one of the di is of
degree zero. In this case we have found the lowest possible D for the degree isomorphism
problem.
Proposition 4.2. Suppose that there exists a j ∈ {1, . . . , m} such that dj is of degree
zero. Then
Q + max (deg(di ))
1≤i≤m
is the lowest possible D for the degree isomorphism problem.
Proof. Without loss of generality assume that d1 is of degree zero. Then we obtain
from Lemma 4.1 that
Gi = deg(gcd(f1 , . . . , fi−1 , fi+1 , . . . , fm ))
(i = 2, . . . , m)
(4.2)
and
G1 = deg(gcd(f2 , f3 , . . . , fm )) + max (deg(dj )).
2≤j≤m
(4.3)
Adding all the Gi ’s gives
m
X
i=1
Gi = Q + max (deg(di )).
2≤i≤m
(4.4)
Since d1 is of degree zero the max in (4.4) can actually be taken over 1 ≤ i ≤ m.
Therefore, it follows from Theorem 2.3 and (4.1) that Q + max1≤i≤m (deg(di )) is the
lowest possible D for the degree isomorphism problem. 2
Theorem 4.3. (Partial Fractions Theorem) All the di are of degree zero if and
only if Q is the lowest possible D in the degree isomorphism problem.
Proof. By (4.1) if Q is the lowest possible D then all di must have degree zero. Conversely, if all the di are of degree zero then by the preceding proposition Q is the lowest
possible D. 2
Low Degree Solutions to Linear Equations with K[x] Coefficients
7
Theorem 4.3 is called the “Partial Fractions Theorem” because Q being the lowest
possible D is equivalent to:
Let q := q1 · · · qm . For every polynomial of
Pdegree lower than q the rational
function b/q has a unique representation
ai /qi , where each ai is of lower
degree than qi .
References
Kakié, K. (1976). The resultant of several homogeneous polynomials in two indeterminates. Proc. AMS
54, 1–7.
Shiffman, B. (1989). Degree bounds for the division problem in polynomial ideals. Michigan Math. J.
36, 163–171.