LPDO: Linear Partial Diﬀerential Operators
Date: 2020-10-07 18:10:00

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### 16.40 LPDO: Linear Partial Diﬀerential Operators

Author: Thomas Sturm

#### 16.40.1 Introduction

Consider the ﬁeld F = ℚ(x1,…,xn) of rational functions and a set Δ = {∂x1,…, ∂xn} of commuting derivations acting on F . That is, for all ∂xi, ∂xj ∈ Δ and all f, g ∈ F the following properties are satisﬁed:

Consider now the set F[∂x1,…,∂xn], where the derivations are used as variables. This set forms a non-commutative linear partial diﬀerential operator ring with pointwise addition, and multiplication deﬁned as follows: For f ∈ F and ∂xi, ∂xj ∈ Δ we have for any g ∈ F that

Here “ ⋅ ” denotes the multiplication in F . From (16.90) and (16.88) it follows that ∂xi∂xj = ∂xj∂xi, and using (16.89) and (16.87) the following commutator can be proved:

A linear partial diﬀerential operator (LPDO) of order k is an element

in canonical form. Here the expression |j|≤ k speciﬁes the set of all tuples of the form j = (j1,…,jn) ∈ ℕn with ∑ i=1nji ≤ k, and we deﬁne ∂j = ∂x1j1⋯∂x njn.

A factorization of D is a non-trivial decomposition

into multiplicative factors, each of which is an LPDO Di of order greater than 0 and less than k. If such a factorization exists, then D is called reducible or factorable, else irreducible.

For the purpose of factorization it is helpful to temporarily consider as regular commutative polynomials certain summands of the LPDO under consideration. Consider a commutative polynomial ring over F in new indeterminates y1, …, yn. Adopting the notational conventions above, for m ≤ k the symbol of D of order m is deﬁned as

For m = k we obtain as a special case the symbol Sym(D) of D.

#### 16.40.2 Operators

##### 16.40.2.1 partial

There is a unary operator partial(⋅) denoting ∂.

| → | partial ( ) |

##### 16.40.2.2 ***

There is a binary operator *** for the non-commutative multiplication involving partials ∂x. All expressions involving *** are implicitly transformed into LPDOs, i.e., into the following normal form:

| → |  |
| → |  |
| → |  |

The summands of the normalized-lpdo are ordered in some canonical way. As an example consider

input: a()***partial(y)***b()***partial(x);
(a()*b()) *** partial(x) *** partial(y) + (a()*diff(b(),y,1)) *** partial(x)

Here the F-elements are polynomials, where the unknowns are of the type constant-operator denoting functions from F :

| → | ( ) |

We do not admit division of such constant operators since we cannot exclude that such a constant operator denotes 0.

The operator notation on the one hand emphasizes the fact that the denoted elements are functions. On the other hand it distinguishes a() from the variable a of a rational function, which speciﬁcally denotes the corresponding projection. Consider e.g.

input: (x+y)***partial(y)***(x-y)***partial(x);
2 2
(x - y ) *** partial(x) *** partial(y) + ( - x - y) *** partial(x)

Here we use as F-elements speciﬁc elements from F = ℚ(x,y).

##### 16.40.2.3 diff

In our example with constant operators, the transformation into normal form introduces a formal derivative operation diff(⋅,⋅,⋅), which cannot be evaluated. Notice that we do not use the Reduce operator df(⋅,⋅,⋅) here, which for technical reasons cannot smoothly handle our constant operators.

In our second example with rational functions as F-elements, derivative occurring with commutation can be computed such that diff does not occur in the output.

#### 16.40.3 Shapes of F-elements

Besides the generic computations with constant operators, we provide a mechanism to globally ﬁx a certain shape for F-elements and to expand constant operators according to that shape.

##### 16.40.3.1 lpdoset

We give an example for a shape that ﬁxes all constant operators to denote generic bivariate aﬃne linear functions:

input: d := (a()+b())***partial(x1)***partial(x2)**2;
2
d := (a() + b()) *** partial(x1) *** partial(x2)
input: lpdoset {!#10*x1+!#01*x2+!#00,x1,x2};
{-1}
input: d;
2
(a00 + a01*x2 + a10*x1 + b00 + b01*x2 + b10*x1) *** partial(x1) *** partial(x2)

Notice that the placeholder # must be escaped with !, which is a general convention for Rlisp/Reduce. Notice that lpdoset returns the old shape and that {-1} denotes the default state that there is no shape selected.

##### 16.40.3.2 lpdoweyl

The command lpdoweyl {n,x1,x2,...} creates a shape for generic polynomials of total degree n in variables x1, x2, ….

input: lpdoweyl(2,x1,x2);
2 2
{#_00_ + #_01_*x2 + #_02_*x2 + #_10_*x1 + #_11_*x1*x2 + #_20_*x1 ,x1,x2}
input: lpdoset ws;
{#10*x1 + #01*x2 + #00,x1,x2}
input: d;
2 2
(a_00_ + a_01_*x2 + a_02_*x2 + a_10_*x1 + a_11_*x1*x2 + a_20_*x1 + b_00_
2 2
+ b_01_*x2 + b_02_*x2 + b_10_*x1 + b_11_*x1*x2 + b_20_*x1 ) *** partial(x1)
2
*** partial(x2)

#### 16.40.4 Commands

##### 16.40.4.1 General

The order of an lpdo:

input: lpdoord((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1));
3

Returns the list of derivations (partials) occurring in its argument LPDO d.

input: lpdoptl(a()***partial(x1)***partial(x2)+partial(x4)+diff(a(),x3,1));
{partial(x1),partial(x2),partial(x4)}

That is the smallest set {…,∂xi,…} such that d is deﬁned in F[…,∂xi,…]. Notice that formal derivatives are not derivations in that sense.

Given a starting symbol a, a list of variables l, and a degree n, lpdogp(a,l,n) generates a generic (commutative) polynomial of degree n in variables l with coeﬃcients generated from the starting symbol a:

input: lpdogp(a,{x1,x2},2);
2 2
a_00_ + a_01_*x2 + a_02_*x2 + a_10_*x1 + a_11_*x1*x2 + a_20_*x1

Given a starting symbol a, a list of variables l, and a degree n, lpdogp(a,l,n) generates a generic diﬀerential polynomial of degree n in variables l with coeﬃcients generated from the starting symbol a:

input: lpdogdp(a,{x1,x2},2);
2 2
a_20_ *** partial(x1) + a_02_ *** partial(x2)
+ a_11_ *** partial(x1) *** partial(x2) + a_10_ *** partial(x1)
+ a_01_ *** partial(x2) + a_00_

##### 16.40.4.2 Symbols

The symbol of an lpdo. That is the diﬀerential monomial of highest order with the partials replaced by corresponding commutative variables:

input: lpdosym((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1));
2
y_x1_*y_x2_ *(a() + b())

More generally, one can use a second optional arguments to specify a the order of a diﬀerent diﬀerential monomial to form the symbol of:

input: lpdosym((a()+b())***partial(x1)***partial(x2)**2+3***partial(x1),1);
3*y_x1_

Finally, a third optional argument can be used to specify an alternative starting symbol for the commutative variable, which is y by default. Altogether, the optional arguments default like lpdosym(⋅)=lpdosym(⋅,lpdoord(⋅),y).

This converts a symbol obtained via lpdosym back into an LPDO resulting in the corresponding diﬀerential monomial of the original LPDO.

input: d := a()***partial(x1)***partial(x2)+partial(x3)$
input: s := lpdosym d;
s := a()*y_x1_*y_x2_
input: lpdosym2dp s;
a() *** partial(x1) *** partial(x2)

In analogy to lpdosym there is an optional argument for specifying an alternative starting symbol for the commutative variable, which is y by default.

Given LPDOs p, q and m ∈ ℕ the function lpdos(p,q,m) computes the commutative polynomial

This is useful for the factorization of LPDOs.
input: p := a()***partial(x1)+b()$
input: q := c()***partial(x1)+d()***partial(x2)$
input: lpdos(p,q,1);
a()*diff(c(),x1,1)*y_x1_ + a()*diff(d(),x1,1)*y_x2_ + b()*c()*y_x1_
+ b()*d()*y_x2_

##### 16.40.4.3 Factorization

Factorize the argument LPDO d. The ground ﬁeld F must be ﬁxed via lpdoset. The result is a list of lists {…,(Ai,Li),…}. Ai is is genrally the identiﬁers true, which indicates reducibility. The respective Li is a list of two diﬀerential polynomial factors, the ﬁrst of which has order 1.

input: bk := (partial(x)+partial(y)+(a10-a01)/2) ***
(partial(x)-partial(y)+(a10+a01)/2);
2 2
bk := partial(x) - partial(y) + a10 *** partial(x) + a01 *** partial(y)
2 2
- a01 + a10
+ ----------------
4
input: lpdoset lpdoweyl(1,x,y);
{#_00_ + #_01_*y + #_10_*x,x,y}
input: lpdofactorize bk;
{{true,
a01 - a10
{ - partial(x) - partial(y) + -----------,
2
- a01 - a10
- partial(x) + partial(y) + --------------}}}
2

If the result is the empty list, then this guarantees that there is no approximate factorization possible. In general it is possible to obtain several sample factorizations. Note, however, that the result does not provide a complete list of possible factorizations with a left factor of order 1 but only at least one such sample factorization in case of reducibility.

Furthermore, the procedure might fail due to polynomial degrees exceeding certain bounds for the extended quantiﬁer elimination by virtual substitution used internally. In this case there is the identiﬁer failed returned. This must not be confused with the empty list indicating irreducibility as described above.

Besides

1. the LPDO d,

lpdofactorizex accepts several optional arguments:

1. An LPDO of order 1, which serves as a template for the left (linear) factor. The default is a generic linear LPDO with generic coeﬃcient functions according from the ground ﬁeld speciﬁed via lpdoset. The principle idea is to support the factorization by guessing that certain diﬀerential monomials are not present.
2. An LPDO of order ord(d)−1, which serves as a template for the right factor. Similarly to the previous argument the default is fully generic.

This is a low-level entry point to the factorization lpdofactorize. It accepts the same arguments as lpdofactorize. It generates factorization conditions as a quite large ﬁrst-order formula over the reals. This can be passed to extended quantiﬁer elimination. For example, consider bk as in the example for lpdofactorize above:

input: faccond := lpdofac bk$
input: rlqea faccond;
{{true,
a01 - a10
{p_00_00_ = -----------,
2
p_00_01_ = 0, p_00_10_ = 0, p_01_00_ = -1, p_01_01_ = 0, p_01_10_ = 0,
p_10_00_ = -1, p_10_01_ = 0, p_10_10_ = 0,
- a01 - a10
q_00_00_ = --------------,
2
q_00_01_ = 0, q_00_10_ = 0, q_01_00_ = 1, q_01_01_ = 0, q_01_10_ = 0,
q_10_00_ = -1, q_10_01_ = 0, q_10_10_ = 0}}}

The result of the extended quantiﬁer elimination provides coeﬃcient values for generic factor polynomials p and q. These are automatically interpreted and converted into diﬀerential polynomials by lpdofactorize.

##### 16.40.4.4 Approximate Factorization

Approximately factorize the argument LPDO d. The ground ﬁeld F must be ﬁxed via lpdoset. The result is a list of lists {…,(Ai,Li),…}. Each Ai is quantiﬁer-free formula possibly containing a variable epsilon, which describes the precision of corresponding factorization Li. Li is a list containing two factors, the ﬁrst of which is linear.

input: off lpdocoeffnorm$
input: lpdoset lpdoweyl(0,x1,x2)$
input: f2 := partial(x1)***partial(x2) + 1$
input: lpdofactorizex f2;
{{epsilon - 1 >= 0,{partial(x1),partial(x2)}},
{epsilon - 1 >= 0,{partial(x2),partial(x1)}}}

If the result is the empty list, then this guarantees that there is no approximate factorization possible. In our example we happen to obtain two possible factorizations. Note, however, that the result in general does not provide a complete list of factorizations with a left factor of order 1 but only at least one such sample factorization.

Furthermore, the procedure might fail due to polynomial degrees exceeding certain bounds for the extended quantiﬁer elimination by virtual substitution used internally. If this happens, the corresponding Ai will contain existential quantiﬁers ex, and Li will be meaningless.
Da sollte besser ein failed kommen ...

The ﬁrst of the two subresults above has the semantics that ∂x1∂x2 is an approximate factorization of f2 for all 𝜀 ≥ 1. Formally, ||f2 − ∂x1∂x2||≤ 𝜀 for all 𝜀 ≥ 1, which is equivalent to ||f2 − ∂x1∂x2||≤ 1. That is, 1 is an upper bound for the approximation error over ℝ2. Where there are two possible choices for the seminorm ||⋅||:

1. ...
2. ...

explain switch lpdocoeffnorm ...

Besides

1. the LPDO d,

lpdofactorizex accepts several optional arguments:

1. A Boolean combination ψ of equations, negated equations, and (possibly strict) ordering constraints. This ψ describes a (semialgebraic) region over which to factorize approximately. The default is true specifying the entire ℝn. It is possible to choose ψ parametrically. Then the parameters will in general occur in the conditions Ai in the result.
2. An LPDO of order 1, which serves as a template for the left (linear) factor, and an LPDO of order ord(d) − 1, which serves as a template for the right factor. See the documentation of lpdofactorize for defaults and details.
3. A bound 𝜀 for describing the desired precision for approximate factorization. The default is the symbol epsilon, i.e., a symbolic choice such that the optimal choice (with respect to parameters in ψ) is obtained during factorization. It is possible to ﬁx 𝜀 ∈ ℚ. This does, however, not considerably simplify the factorization process in most cases.

input: f3 := partial(x1) *** partial(x2) + x1$
input: psi1 := 0<=x1<=1 and 0<=x2<=1$
input: lpdofactorizex(f3,psi1,a()***partial(x1),b()***partial(x2));
{{epsilon - 1 >= 0,{partial(x1),partial(x2)}}}

This is a low-level entry point to the factorization lpdofactorizex. It is analogous to lpdofac for lpdofactorize; see the documentation there for details.

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