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\( \newcommand {\Sym }{\operatorname {Sym}} \)Consider the field \(F = \mathbb {Q}(x_1,\dots ,x_n)\) of rational functions and a set \(\Delta = \{\partial _{x_1}\),…, \(\partial _{x_n}\}\) of commuting derivations acting on \(F\). That is, for all \(\partial _{x_i}\), \(\partial _{x_j} \in \Delta \) and all \(f\), \(g\in F\) the following properties are satisfied:
Consider now the set \(F[\partial _{x_1},\dots ,\partial _{x_n}]\), 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 \(f\in F\) and \(\partial _{x_i}\), \(\partial _{x_j} \in \Delta \) we have for any \(g\in F\) that
Here “\({}\cdot {}\)” denotes the multiplication in \(F\). From (\ref {EQ:mult3}) and (\ref {EQ:comm}) it follows that \(\partial _{x_i} \partial _{x_j}=\partial _{x_j} \partial _{x_i}\), and using (\ref {EQ:mult2}) and (\ref {EQ:leibnitz}) the following commutator can be proved:
A linear partial differential operator (LPDO) of order \(k\) is an element
A factorization of \(D\) is a non-trivial decomposition
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 \(y_1\), …, \(y_n\). Adopting the notational conventions above, for \(m\leq k\) the symbol of \(D\) of order \(m\) is defined as
partial There is a unary operator partial(\(\cdot \)) denoting \(\partial \).
| \(\langle \)partial-term\(\rangle \) | \(\ \ \rightarrow \ \ \) | partial ( \(\langle \)id\(\rangle \) ) | |
*** There is a binary operator *** for the non-commutative multiplication involving
partials \(\partial _x\). All expressions involving *** are implicitly transformed into LPDOs, i.e., into
the following normal form:
| \(\langle \)normalized-lpdo\(\rangle \) | \(\ \ \rightarrow \ \ \) | \(\langle \)normalized-mon\(\rangle \) \([\;\)+ \(\langle \)normalized-lpdo\(\rangle \) \(\;]\) | |
| \(\langle \)normalized-mon\(\rangle \) | \(\ \ \rightarrow \ \ \) | \(\langle \)F-element\(\rangle \) \([\;\)*** \(\langle \)partial-termprod\(\rangle \) \(\;]\) | |
| \(\langle \)partial-termprod\(\rangle \) | \(\ \ \rightarrow \ \ \) | \(\langle \)partial-term\(\rangle \) \([\;\)*** \(\langle \)partial-termprod\(\rangle \) \(\;]\) | |
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 \(F\):
| \(\langle \)constant-operator\(\rangle \) | \(\ \ \rightarrow \ \ \) | \(\langle \)id\(\rangle \) ( ) | |
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 \(F=\mathbb {Q}(x,y)\).
diff In our example with constant operators, the transformation into normal form introduces
a formal derivative operation diff(\(\cdot \),\(\cdot \),\(\cdot \)), which cannot be evaluated. Notice that we do
not use the Reduce operator df(\(\cdot \),\(\cdot \),\(\cdot \)) 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.
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.
lpdosetWe 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;
(a00 + a01*x2 + a10*x1 + b00 + b01*x2 + b10*x1)
2
*** 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.
lpdoweylThe command lpdoweyl {n,x1,x2,...} creates a shape for generic polynomials
of total degree n in variables \(\texttt {x1}\), \(\texttt {x2}\), ….
input: lpdoweyl(2,x1,x2);
2
{#_00_ + #_01_*x2 + #_02_*x2 + #_10_*x1
2
+ #_11_*x1*x2 + #_20_*x1 ,x1,x2}
input: lpdoset ws;
{#10*x1 + #01*x2 + #00,x1,x2}
input: d;
2
(a_00_ + a_01_*x2 + a_02_*x2 + a_10_*x1
2
+ a_11_*x1*x2 + a_20_*x1 + b_00_ + b_01_*x2
2 2
+ b_02_*x2 + b_10_*x1 + b_11_*x1*x2 + b_20_*x1
2
) *** partial(x1) *** partial(x2)
lpdoordinput: lpdoord((a()+b())***partial(x1) ***partial(x2)**2+3***partial(x1)); 3
lpdoptlReturns 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 \(\{\dots ,\partial _{x_i},\dots \}\) such that \(d\) is defined in \(F[\dots ,\partial _{x_i},\dots ]\). Notice that formal derivatives are not derivations in that sense.
lpdogpGiven 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
lpdogdpGiven 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_
lpdosymThe 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(\(\cdot \))=lpdosym(\(\cdot \),lpdoord(\(\cdot \)),y).
lpdosym2dpThis 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.
lpdosGiven LPDOs \(p\), \(q\) and \(m\in \mathbb {N}\) the function lpdos(\(p\),\(q\),\(m\)) computes the commutative polynomial
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_
lpdofactorizeFactorize the argument LPDO \(d\). The ground field \(F\) must be fixed via lpdoset. The result
is a list of lists \(\{\dots ,(A_i,L_i),\dots \}\). \(A_i\) is is genrally the identifiers true, which indicates reducibility. The
respective \(L_i\) 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)
2 2
- a01 + a10
+ a01*partial(y) + ----------------
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
lpdofactorize accepts several optional arguments:
lpdoset. The principle idea
is to support the factorization by guessing that certain differential monomials
are not present.
lpdofacThis 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.
lpdofactorizexApproximately factorize the argument LPDO \(d\). The ground field \(F\) must be fixed via
lpdoset. The result is a list of lists \(\{\dots ,(A_i,L_i),\dots \}\). Each \(A_i\) is quantifier-free formula possibly
containing a variable \(\texttt {epsilon}\), which describes the precision of corresponding factorization \(L_i\). \(L_i\) 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 \(A_i\) will contain existential quantifiers ex, and \(L_i\) will be
meaningless.Da sollte besser ein failed kommen ...
The first of the two subresults above has the semantics that \(\partial _{x_1}\partial _{x_2}\) is an approximate factorization of \(f_2\) for all \(\varepsilon \geq 1\). Formally, \(||f_2-\partial _{x_1}\partial _{x_2}||\leq \varepsilon \) for all \(\varepsilon \geq 1\), which is equivalent to \(||f_2-\partial _{x_1}\partial _{x_2}||\leq 1\). That is, \(1\) is an upper bound for the approximation error over \(\mathbb {R}^2\). Where there are two possible choices for the seminorm \(||\cdot ||\):
explain switch lpdocoeffnorm ...
Besides
lpdofactorizex accepts several optional arguments:
true specifying the entire
\(\mathbb {R}^n\). It is possible to choose \(\psi \) parametrically. Then the parameters will in general
occur in the conditions \(A_i\) in the result.
lpdofactorize for defaults and details.
epsilon, i.e., a symbolic choice such that
the optimal choice (with respect to parameters in \(\psi \)) is obtained during
factorization. It is possible to fix \(\varepsilon \in \mathbb {Q}\). 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)}}}
lpdofacxThis is a low-level entry point to the factorization lpdofactorizex. It is
analogous to lpdofac for lpdofactorize; see the documentation there for
details.
lpdohrect
lpdohcirc
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