# Loewy decomposition

In the study of differential equations, the Loewy decomposition breaks every linear ordinary differential equation (ODE) into what are called largest completely reducible components. It was introduced by Alfred Loewy.

Solving differential equations is one of the most important subfields in mathematics. Of particular interest are solutions in closed form. Breaking ODEs into largest irreducible components, reduces the process of solving the original equation to solving irreducible equations of lowest possible order. This procedure is algorithmic, so that the best possible answer for solving a reducible equation is guaranteed. A detailed discussion may be found in.

Loewy's results have been extended to linear partial differential equations (PDEs) in two independent variables. In this way, algorithmic methods for solving large classes of linear pde's have become available.

## Decomposing linear ordinary differential equations

$L\equiv D^{n}+a_{1}D^{n-1}+\cdots +a_{n-1}D+a_{n}$ For any two operators $L_{1}$ and $L_{2}$ the least common left multiple $Lclm(L_{1},L_{2})$ is the operator of lowest order such that both $L_{1}$ and $L_{2}$ divide it from the right. The greatest common right divisior $Gcrd(L_{1},L_{2})$ is the operator of highest order that divides both $L_{1}$ and $L_{2}$ from the right. If an operator may be represented as $Lclm$ of irreducible operators it is called completely reducible. By definition, an irreducible operator is called completely reducible.

If an operator is not completely reducible, the $Lclm$ of its irreducible right factors is divided out and the same procedure is repeated with the quotient. Due to the lowering of order in each step, this proceeding terminates after a finite number of iterations and the desired decomposition is obtained. Based on these considerations, Loewy  obtained the following fundamental result.

$L\equiv D^{n}+a_{1}D^{n-1}+\cdots +a_{n-1}D+a_{n}$ $L=L_{m}^{(d_{m})}L_{m-1}^{(d_{m-1})}\ldots L_{1}^{(d_{1})}$ $L_{k}^{(d_{k})}=Lclm(l_{j_{1}}^{(e_{1})},l_{j_{2}}^{(e_{2})},\ldots ,l_{j_{k}}^{(e_{k})})$ The decomposition determined in this theorem is called the Loewy decomposition of $L$ . It provides a detailed description of the function space containing the solution of a reducible linear differential equation $Ly=0$ .

For operators of fixed order the possible Loewy decompositions, differing by the number and the order of factors, may be listed explicitly; some of the factors may contain parameters. Each alternative is called a type of Loewy decomposition. The complete answer for $n=2$ is detailed in the following corollary to the above theorem.

${\mathcal {L}}_{1}^{2}:L=l_{2}^{(1)}l_{1}^{(1)};$ ${\mathcal {L}}_{2}^{2}:L=Lclm(l_{2}^{(1)},l_{1}^{(1)});$ ${\mathcal {L}}_{3}^{2}:L=Lclm(l^{(1)}(C)).$ The decomposition type of an operator is the decomposition ${\mathcal {L}}_{i}^{2}$ with the highest value of $i$ . An irreducible second-order operator is defined to have decomposition type ${\mathcal {L}}_{0}^{2}$ .

${\mathcal {L}}_{1}^{2}$ : $Ly=(D+a_{2})(D+a_{1})y=0$ ;
$y_{2}=\varepsilon _{1}(x)\int {\frac {\varepsilon _{2}(x)}{\varepsilon _{1}(x)}}\,dx.$ ${\mathcal {L}}_{2}^{2}$ : $Ly=Lclm(D+a_{2},D+a_{1})y=0;$ $y_{i}=\varepsilon _{i}(x);$ ${\mathcal {L}}_{3}^{2}$ : $Ly=Lclm(D+a(C))y=0;$ $y_{1}=\varepsilon (x,{\bar {C}})$ }
$y_{2}=\varepsilon (x,{\bar {\bar {C}}}).$ Here two rational functions $p,q\in {\mathbb {Q} }(x)$ are called equivalent if there exists another rational function $r\in {\mathbb {Q} }(x)$ such that

$p-q={\frac {r'}{r}}$ .

There remains the question how to obtain a factorization for a given equation or operator. It turns out that for linear ode's finding the factors comes down to determining rational solutions of Riccati equations or linear ode's; both may be determined algorithmically. The two examples below show how the above corollary is applied.

Example 1 Equation 2.201 from Kamke's collection. has the ${\mathcal {L}}_{2}^{2}$ decomposition

$y_{1}={\frac {2}{3}}-{\frac {4}{3x}}+{\frac {1}{x^{2}}},$ $y_{2}={\frac {2}{x}}+{\frac {3}{x^{2}}}e^{-2x}.$ Example 2 An equation with a type ${\mathcal {L}}_{3}^{2}$ decomposition is

$y''-{\frac {6}{x^{2}}}y=Lclm{\big (}D+{\frac {2}{x}}-{\frac {5x^{4}}{x^{5}+C}}{\big )}y=0.$ These results show that factorization provides an algorithmic scheme for solving reducible linear ode's. Whenever an equation of order 2 factorizes according to one of the types defined above the elements of a fundamental system are explicitly known, i.e. factorization is equivalent to solving it.

A similar scheme may be set up for linear ode's of any order, although the number of alternatives grows considerably with the order; for order $n=3$ the answer is given in full detail in.

If an equation is irreducible it may occur that its Galois group is nontrivial, then algebraic solutions may exist. If the Galois group is trivial it may be possible to express the solutions in terms of special function like e.g. Bessel or Legendre functions, see  or.

## Basic facts from differential algebra

In order to generalize Loewy's result to linear pde's it is necessary to apply the more general setting of differential algebra. Therefore a few basic concepts that are required for this purpose are given next.

A field ${\mathcal {F}}$ is called a differential field if it is equipped with a derivation operator. An operator $\delta$ on a field ${\mathcal {F}}$ is called a derivation operator if $\delta (a+b)=\delta (a)+\delta (b)$ and $\delta (ab)=\delta (a)b+a\delta (b)$ for all elements $a,b\in {\mathcal {F}}$ . A field with a single derivation operator is called an ordinary differential field; if there is a finite set containing several commuting derivation operators the field is called a partial differential field.

The generators of an ideal are highly non-unique; its members may be transformed in infinitely many ways by taking linear combinations of them or its derivatives without changing the ideal. Therefore M. Janet  introduced a normal form for systems of linear pde's that has been baptized Janet basis. They are the differential analog to Groebner bases of commutative algebra, originally they have been introduced by Bruno Buchberger; therefore they are also called differential Groebner basis.

In order to generate a Janet basis, a ranking of derivatives must be defined. It is a total ordering such that for any derivatives $\delta$ , $\delta _{1}$ and $\delta _{2}$ , and any derivation operator $\theta$ the relations $\delta \preceq \theta \delta$ , and $\delta _{1}\preceq \delta _{2}\rightarrow \delta \delta _{1}\preceq \delta \delta _{2}$ are valid. Here graded lexicographic term orderings $grlex$ are applied. For partial derivatives of a single function their definition is analogous to the monomial orderings in commutative algebra. The S-pairs in commutative algebra correspond to the integrability conditions.

Example 3 Consider the ideal

 $I={\Big \langle }l_{1}\equiv \partial _{xx}-{\frac {1}{x}}\partial _{x}-{\frac {y}{x(x+y)}}\partial _{y},$ $l_{1,y}=l_{2,x}-l_{2,y}={\frac {y+2x}{x(x+y)}}\partial _{xy}+{\frac {y}{x(x+y)}}\partial _{yy}$ is reduced w.r.t. to $I$ , the new generator $\partial _{y}$ is obtained. Adding it to the generators and performing all possible reductions, the given ideal is represented as $I={{\Big \langle }{\Big \langle }}\partial _{xx}-{\frac {1}{x}}\partial _{x},\partial _{y}{{\Big \rangle }{\Big \rangle }}$ . Its generators are autoreduced and the single integrability condition is satisfied, i.e. they form a Janet basis.

Given any ideal $I$ it may occur that it is properly contained in some larger ideal $J$ with coefficients in the base field of $I$ ; then $J$ is called a divisor of $I$ . In general, a divisor in a ring of partial differential operators need not be principal.

The least common left multiple (Lclm) or left intersection of two ideals $I$ and $J$ is the largest ideal with the property that it is contained both in $I$ and $J$ . The solution space of $Lclm(I,J)z=0$ is the smallest space containing the solution spaces of its arguments.

A special kind of divisor is the so-called Laplace divisor of a given operator $L$ , page 34. It is defined as follows.

Definition Let $L$ be a partial differential operator in the plane; define

${\mathfrak {l}}_{m}\equiv \partial _{x^{m}}+a_{m-1}\partial _{x^{m-1}}+\ldots +a_{1}\partial _{x}+a_{0}$ and

${\mathfrak {k}}_{n}\equiv \partial _{y^{n}}+b_{n-1}\partial _{y^{n-1}}+\ldots +b_{1}\partial _{y}+b_{0}$ In order for a Laplace divisor to exist the coeffients of an operator $L$ must obey certain constraints. An algorithm for determining an upper bound for a Laplace divisor is not known at present, therefore in general the existence of a Laplace divisor may be undecidable

## Decomposing second-order linear partial differential equations in the plane

Applying the above concepts Loewy's theory may be generalized to linear pde's. Here it is applied to individual linear pde's of second order in the plane with coordinates $x$ and $y$ , and the principal ideals generated by the corresponding operators.

Second-order equations have been considered extensively in the literature of the 19th century,. Usually equations with leading derivatives $\partial _{xx}$ or $\partial _{xy}$ are distinguished. Their general solutions contain not only constants but undetermined functions of varying numbers of arguments; determining them is part of the solution procedure. For equations with leading derivative $\partial _{xx}$ Loewy's results may be generalized as follows.

Theorem 2 Let the differential operator $L$ be defined by

  $L\equiv \partial _{xx}+A_{1}\partial _{xy}+A_{2}\partial _{yy}+A_{3}\partial _{x}+A_{4}\partial _{y}+A_{5}$ where $A_{i}\in {\mathbb {Q} }(x,y)$ for all $i$ .


In order to apply this result for solving any given differential equation involving the operator $L$ the question arises whether its first-order factors may be determined algorithmically. The subsequent corollary provides the answer for factors with coefficients either in the base field or a universal field extension.

Corollary 3 In general, first-order right factors of a linear pde in the base field cannot be determined algorithmically. If the symbol polynomial is separable any factor may be determined. If it has a double root in general it is not possible to determine the right factors in the base field. The existence of factors in a universal field, i.e. absolute irreducibility, may always be decided.

The above theorem may be applied for solving reducible equations in closed form. Because there are only principal divisors involved the answer is similar as for ordinary second-order equations.

Proposition 1 Let a reducible second-order equation

 $Lz\equiv z_{xx}+A_{1}z_{xy}+A_{2}z_{yy}+A_{3}z_{x}+A_{4}z_{y}+A_{5}z=0$ where $A_{1},\ldots ,A_{5}\in {\mathbb {Q} }(x,y)$ .


A differential fundamental system has the following structure for the various decompositions into first-order components.

A typical example of a linear pde where factorization applies is an equation that has been discussed by Forsyth , vol. VI, page 16,

Example 5 (Forsyth 1906)} Consider the differential equation $z_{xx}-z_{yy}+{\frac {4}{x+y}}z_{x}=0$ . Upon factorization the representation

Consequently a differential fundamental system is

If the only second-order derivative of an operator is $\partial _{xy}$ , its possible decompositions involving only principal divisors may be described as follows.

Theorem 3 Let the differential operator $L$ be defined by

In addition there are five more possible decomposition types involving non-principal Laplace divisors as shown next.

Theorem 4 Let the differential operator $L$ be defined by

An equation that does not allow a decomposition involving principal divisors but is completely reducible w.r.t. non-principal Laplace divisors of type ${\mathcal {L}}_{xy}^{4}$ has been considered by Forsyth.

Example 6 (Forsyth 1906) Define

generating the principal ideal $\langle L\rangle$ . A first-order factor does not exist. However, there are Laplace divisors

${\mathbb {L} }_{x^{2}}(L)\equiv {{\Big \langle }{\Big \langle }}\partial _{xx}-{\frac {2}{x-y}}\partial _{x}+{\frac {2}{(x-y)^{2}}},L{{\Big \rangle }{\Big \rangle }}$ and ${\mathbb {L} }_{y^{2}}(L)\equiv {{\Big \langle }{\Big \langle }}L,\partial _{yy}+{\frac {2}{x-y}}\partial _{y}+{\frac {2}{(x-y)^{2}}}{{\Big \rangle }{\Big \rangle }}.$ $z_{1}(x,y)=2(x-y)F(y)+(x-y)^{2}F'(y)$ and $z_{2}(x,y)=2(y-x)G(x)+(y-x)^{2}G'(x)$ .


## Decomposing linear pde's of order higher than 2

It turns out that operators of higher order have more complicated decompositions and there are more alternatives, many of them in terms of non-principal divisors. The solutions of the corresponding equations get more complex. For equations of order three in the plane a fairly complete answer may be found in. A typical example of a third-order equation that is also of historical interest is due to Blumberg .

Example 7 (Blumberg 1912) In his dissertation Blumberg considered the third order operator

it may be written as $Lclm(l_{2},l_{1})={\langle \langle }L_{1},L_{2}{\rangle \rangle }$ . Consequently the Loewy decomposition of Blumbergs's operator is

It yields the following differential fundamental system for the differential equation $Lz=0$ .

Factorizations and Loewy decompositions turned out to be an extremely useful method for determining solutions of linear differential equations in closed form, both for ordinary and partial equations. It should be possible to generalize these methods to equations of higher order, equations in more variables and system of differential equations.