Algebraic solutions of linear differential equations: An arithmetic approach
Abstract
Given a linear differential equation with coefficients in an important question is to know whether its full space of solutions consists of algebraic functions, or at least if one of its specific solutions is algebraic. After presenting motivating examples coming from various branches of mathematics, we advertise in an elementary way a beautiful local-global arithmetic approach to these questions, initiated by Grothendieck in the late sixties. This approach has deep ramifications and leads to the still unsolved Grothendieck–Katz , conjecture. -curvature
1. Context, motivation, and basic examples
In this text we consider linear differential equations (LDEs) of order
where the are known rational functions in with , not identically zero and is an unknown “function”. In many applications, the desired solution is a formal power series with coefficients in Therefore, in what follows, when we write “function” we actually mean an element of . unless otherwise specified. We will say that a function is differentially finite (in short, D-finite) if it satisfies a linear differential equation like Equation 1.
A function is called algebraic if it is algebraic over that is, if , satisfies a polynomial equation of the form for some , Otherwise, . is called transcendental. The simplest algebraic functions are polynomials in closely followed by rational power series: these are rational functions in , that have no pole at and therefore admit a Taylor expansion around the origin. A little more general are roots of rational power series, such as th In all these three cases, . is clearly D-finite and satisfies a linear differential equation of order .
Many other examples of interesting functions that might or might not be solutions of linear differential equations arise from combinatorics. A basic example is given by the Catalan numbers.
By definition, a Dyck path is a path drawn in the quarter plane that starts at consists of steps , (directed by the vector or ) (directed by the vector and finally ends on the ) (see Figure -axis1).
Let be the number of Dyck paths ending at we say that such paths have length ; For instance, . since there is a single Dyck path ending at namely , – while , since there are two Dyck paths ending at namely , – – – and – – – We use the convention that . We notice that any Dyck path of length . can be written uniquely as the concatenation of
- (1)
a step ,
- (2)
a Dyck path of length (translated by ),
- (3)
a step and
- (4)
a Dyck path of length .
It follows that the sequence satisfies the following nonlinear recurrence relation:
If denotes the generating function of the i.e., , the previous relation translates to the algebraic identity ,
(the summand comes from the fact that Therefore, ). is algebraic and one can even solve equation Equation 2 and get the closed formula It is worth noting that, starting from the algebraic relation .Equation 2, one can also derive a linear differential equation satisfied by Indeed, differentiating .Equation 2, one gets Therefore, .
The right-hand side in the latter expression can be further simplified using equation Equation 2 again. Notice that
and, consequently,
after replacing two times by .
Finally, one obtains the inhomogeneous differential equation
From this, we can derive new interesting information about the sequence For instance, it easily implies the simpler recurrence relation . for all from which we further derive the closed formula , Using Stirling’s formula, we also deduce the asymptotic estimate . .
The previous example shows that being able to write down an equation (either algebraic or differential) for a generating series can help a lot in studying its coefficients. Of course, obtaining explicit closed formulas (as we did for the Catalan numbers) will not be possible in general; however, meaningful information (such as the asymptotic growth of the coefficients) can often be extracted from the equation. Besides, in many cases it turns out that the algebraicity of a generating series is the mirror of a (sometimes hidden) “algebraic” structure on the combinatorial side, which often takes the form of a recursive tree structure: in Example 1.1, for instance, a Dyck path can be decomposed as a concatenation of smaller Dyck paths, which themselves can be decomposed similarly, etc. We refer to
In Example 1.1, we transformed an algebraic equation into a differential equation. It is actually a general fact and an old result, already known by Abel, that any algebraic function is D-finite. Precisely if satisfies an algebraic equation with of degree in then , also satisfies a differential equation like Equation 1 of order bounded from above by This follows easily from the following reasoning. By differentiating . with respect to and by using the chain rule, we obtain the equality
Here and in what follows, we denote by the derivative of with respect to Therefore, if . is assumed to be a polynomial of minimal degree in satisfied by then , is a nonzero function in and hence , is a rational function in By using the equation . again, it is easy to see that any rational function in can be re-written as a polynomial of degree at most in In other terms, the derivative . lives in the space generated by -vector Similarly, the same holds for all derivatives . and hence these elements must satisfy a nontrivial linear relation over , any such relation yields a linear differential equation ;Equation 1 of order at most Observe that the same reasoning also proves the existence of an inhomogeneous linear differential equation of order at most . for .
A naive though very natural question is whether the converse of Abel’s result holds: is every D-finite function algebraic? The answer is negative, already for differential equations of order as the following example shows. ,
The function solution of , is transcendental. Here is a purely algebraic proof. Let us assume by contradiction that , satisfies a polynomial equation, of minimal degree of the form , for some rational functions By differentiating this equality with respect to . and using we get a new ,degree- equation which, by minimality, is equal the former up to a factor , In other words, . for all In particular, . which implies that , (Indeed, if . for two coprime polynomials with then , divides hence , divides and Thus . which implies , and The nullity of .) now implies that satisfies a polynomial equation of degree which contradicts the minimality of , .
The reader could object that in Example 1.2 we were probably lucky, because the differential equation of is so simple, being of order 1 with constant coefficients. Indeed, in the particular case of the exponential function, there are many other ad hoc transcendence proofs, based on various branches of mathematics. For instance, a direct analytic argument is that, viewed as a complex analytic function, any nonpolynomial algebraic function needs to have a finite (and positive) radius of convergence, while is entire (that is, analytic in the whole complex plane). Another proof is that cannot satisfy a nontrivial algebraic equation, since by otherwise specializing that equation at we would find that the number is an algebraic number, a statement known be to false since Hermite (1873). One could qualify this last proof as “cheating”, since it is intuitively clear that proving transcendence of functions should be easier than proving transcendence of numbers.
A systematic and very useful analytic way to establish functional transcendence is Flajolet’s criterion
where the variable is a complex number with positive real part. The gamma function interpolates the factorial in the sense that for any positive integer .
Let be an algebraic nonpolynomial function. Then has a finite number of singularities, a finite nonzero radius of convergence, and its coefficient sequence is such that
where , , , and ,
The most useful form of the criterion is Corollary 1.4.
If
As an example of application, Proposition 1.3 immediately implies that
At this point, we can ask ourselves: is there a purely arithmetic proof of the transcendence of
Is there any number-theoretic way to recognize whether the differential equation Equation 1 admits only algebraic solutions in its solution space?
Nicely enough, the answer to this question is positive, for two distinct but related reasons. Let us first explain them a bit in the case of the exponential function
If the function
is algebraic, then there exists
Since in the factorial sequence
To formulate the second arithmetic proof of the transcendence of
We denote by
Although the ring
The ring
Using these notions, we can now formulate a very basic but important arithmetic result.
If all solutions of Equation 1 are algebraic functions, then for all but a finite number of prime numbers
By Eisenstein’s criterion (Proposition 1.6), there exists a basis of algebraic solutions
The generating function of the Catalan numbers