Unsolved Problems

Here and further on we deal with holomorphic functions and their derivatives in the complex plane.

The Conjecture (a brief presentation here (slides 6-8),  in details here) is not the only unsolved problem emerging in the frame of the Unifying View, but it is the central open issue affecting the shape of the Unifying View theory. Below are a few problems topped by the Conjecture proper and including other unsolved problems associated with the Conjecture, all awaiting attention of the mathematical community.

1) The Conjecture as a gap in the Unifying View and the foundation of Automatic Differentiation

In either of the two equivalent forms below, the Conjecture claims that ...

 For any component, say x(t), of an explicit system in x(t), y(t), z(t), ... of m 1st order rational ODEs  x'(t) = r1(t, x, y, z, ...) y'(t) = r2(t, x, y, z, ...)                     (1) z'(t) = r3(t, x, y, z, ...) . . . . . . . . . . .   regular at an initial point t=a For any component, say x(t), of an explicit system in x(t), y(t), z(t), ... of m 1st order polynomial ODEs(1) x'(t) = p1(t, x, y, z, ...) y'(t) = p2(t, x, y, z, ...)                   (2) z'(t) = p3(t, x, y, z, ...) . . . . . . . . . . . regular at all points of its phase space, for any initial point a there exists an explicit n-order rational ODE satisfied by x(t)   x(n) = R(t, x, x', x", ..., x(n-1))              (3) or an implicit polynomial ODE Q(t, x, x', x", ..., x(n)) = 0                    (4) both regular at the same initial point t=a  meaning that the denominator g of R=f/g  is nonzero at a, or ∂Q/∂Xn|t=a ≠ 0  (where Xk  denotes x(k)).

So far neither the proof nor counterexample for this Conjecture is available. However if the Conjecture is weakened by dropping the requirement of regularity in the target ODE (3), there is a proof based on combinatorial considerations (though impractical for obtaining the target ODE) – see the slide 13 or Appendix 1 here) .
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(1) 1st order rational ODEs regular at a given point are transformable into a wider polynomial system of ODEs.

Evolution of the concept of elementary functions

First elementary functions were defined by Liouville in the 19th century as a matter of convention: a few well studied vastly used functions and their finite compositions.

In the 1960s Ramon Moore suggested a different approach, replacing the convention about the elementary functions with a definition based on their fundamental property.

Moore defined (vector-) elementariness based on a property of an elementary vector-function to satisfy a system of explicit rational 1st order ODEs.

A competing definition of stand-alone or scalar elementariness would be based on a property of an elementary function to satisfy one explicit rational ODE of order n, or one implicit polynomial ODE of order n.

The vector elementariness by Moore easily follows from the scalar elementariness (because conversion of an n-order ODE into a system of 1st order ODEs is trivial). As to the vice versa statement, the scalar elementariness by Moore follows from vector elementariness due to the Theorem proved by me in Appendix 1 here. Therefore the scalar and vector elementariness by Moore are equivalent.

The Liouville-elementary functions are also Moore-elementary. Say cos t is vector-elementary together with sin t because they satisfy the system

x' = -y
y' = x .

However each of them is also scalar-elementary satisfying the ODE  x" = x (or satisfying an
implicit first order non-linear ODE  (x')2 x2 = 1). In general case however it may be a challenge to obtain an n-order ODE demonstrating the scalar elementariness from the system of ODEs demonstrating the vector-elementariness.

It was understood indeed that there exist infinitely many ODEs for demonstrating elementariness of a particular (vector-) function: some ODEs possibly singular at certain points of their phase space.

It's worth noting that...

The Moore definition applies to the functions...
• In their entire domain of existence, and ...

• With understanding that the rational ODEs may have points of singularity in their phase space unusable as initial points, though sometimes it's possible to replace singular ODEs with the regular ones at the point of interest.

Then in 2008 it was discovered that there exist isolated points in some holomorphic scalar-elementary functions whose Moore-elementariness at such points may be demonstrated only with singular ODEs. For example such is the function

 x(t) = et – 1 t
x(0) = 1

considered in details below. Such kind of special points were earlier known as "regular singularities" or "removable singularities". As it was shown, though the singularity in functions at such point is removable and the functions are holomorphic (with the proper defined value at such point), it's impossible to rid of singularity in the ODEs demonstrating their Moore-like scalar-elementariness at such points. This new type of specialty happens to be unremovable (discussed in item 2).

It was namely the discovery of these special points which gave the reason to refine the Moore definition of elementariness as a property of a function at a neighborhood of a point (rather than a property in the entire domain of existence).

The refined definitions of elementariness

Definition 1: a vector-function (x, y, z, ...) is called vector-elementary at and near a point  t=a  if it satisfies a system of rational ODEs (1) regular at the point t=a, or a polynomial system (2).

Definition 2: a function x(t) is called scalar-elementary at and near a point  t=a  if it satisfies a rational ODE (3) regular at the point t=a, or if it satisfies an implicit polynomial ODE (4) regular at a meaning P/Xm|t=a  0  (where Xm  denotes x(m)).

The vector elementariness easily follows from the scalar elementariness (because conversion of an n-order ODE into a system of 1st order ODEs is trivial). As to the vice versa statement, the scalar elementariness would follow from the vector elementariness if the Conjecture is true. Therefore equivalence between these two definitions depends on the Conjecture.

Most properties and results in the "Unifying View" are proved only for vector-elementariness, and it is not known whether they are true for scalar elementariness. In particular, the fundamental property of closeness of the class of vector-elementary functions states that a composition of such vector-functions, or the inverse vector-function preserves vector-elementariness. (For the scalar elementariness this is not known).

2) How to establish that (scalar) elementariness of a function is violated at a certain point?

Until recently the only function proved to be non-elementary (at all points of its domain!) was the Euler's Gamma function – thanks to the Hölder theorem utilizing the special property of the Gamma function to satisfy the finite difference equation Γ(x+1) = xΓ(x).

Our method of proof that scalar elementariness of a function is violated at a certain point utilizes the specific of the sequence of derivatives at this point, for example the specific of
x(n)|t=0 = 1/(n+1) (see the ).

Any holomorphic function x(t) is defined entirely by its analytic element, i.e. by the sequence of derivatives x(n)|t=a at a point. It appears that the scalar elementariness at a point a also depends on the property of the sequence of its derivatives x(n)|t=a . For the type of functions and examples considered in the article, the sequences of derivatives are rational numbers x(n)|t=a =rn .

The method of proof is based on considering polynomial ODEs   P(t, X, X1,…, Xm)=0  over integer coefficients at a regular point where P/Xm =c 0. By applying differentiation, we may obtain an infinite sequence of polynomial ODEs  Pk(t, X, X1,…, Xm,…, Xm+k)=0  and then to establish that for all leading derivatives  P/Xm+k =c 0.

As a first step, the study was made for a particular function

 x(t)|t=0 = et – 1 t

where all derivatives exist and   x(n)|t=0= 1/(n+1). The idea of the proof was in that while k grows to infinity, the denominator 1/(n+k) passes through all prime numbers. Therefore, for a big enough prime n+k, the fraction 1/(n+k) in the equation Pk =0 cannot cancel with other fractions, leading to a contradiction. That way it was proven that x(t) at t=0  can satisfy no regular polynomial ODE, meaning that its scalar elementariness is violated at  t=0 – followed merely from the fact that the sequence of derivatives x(n)|t=a are fractions (in lower terms) with growing denominators!

This observation enabled uncovering a wide class of functions whose sequence of derivatives is also represented by fractions with infinitely growing denominator (see Table 1 in the article), and here are examples of the respective functions all having t=0 as a point of violation of scalar elementariness:

 et – 1 sin t ln(t + 1) cos t½ t t t

It's notable that (sin t)/t at t=0 represents the "remarkable limit" (studied in the foundation of the calculus). It's new meaning is that it is a point of violation of the scalar elementariness of the respective entire function. The function (ln (1+t))/t is also closely related to the "remarkable limit" (1+t)1/t (see section 4).

And here comes an unsolved part. There are functions – suspects for having points of violation of elementariness, whose sequence of fractions representing the derivatives at the point in question does not fall into the above pattern of fractions with denominator passing throw all prime numbers. For example, the family of Bessel functions J(t) satisfies the known ODE  t2x" + tx'+ (t2 – p2)x=0  singular at  t=0, and it is not known whether it satisfies a regular ODE at t=0. For every function of the Bessel family Jp the expansion at  t=0  is known and its general term is

 Jp(n) = (-1)kCkp+2k 2p+2k

for   n=p+2k,  k=0, 1, 2,…, and zero otherwise. However it's impossible to draw a controversy from these fractions following the method of proof above (the denominators are not growing primes).

Do the Bessel functions have violation of scalar elementariness at t=0?

What about points of violation of vector elementariness?

Another unsolved part of the problem is whether the examples of functions above having t=0 as a point of violation of their scalar elementariness violate also their vector elementariness at that point (the answer depending on the equivalency of the two definitions, or on the Conjecture).

3) A question about the functions having no points of violation
of scalar elementariness

At point  t=0 all the functions above can satisfy no regular rational ODE – and consequently no explicit polynomial ODE; but which kind of ODEs can they satisfy at points other than  t=0?

Indeed, at points other than  t=0  they do satisfy rational ODEs (or implicit polynomial ODEs). As to explicit polynomial ODEs,  here is the

Theorem 1. If a function x(t)  has a point of violation of its scalar elementariness, it can satisfy no explicit polynomial ODE

x(n) = P(t, x, x', x", ..., x(n-1))                           (5)

at either point of its phase space.

The equivalent form of this Theorem is...

Theorem 2. If a function x(t) satisfies an explicit polynomial ODE (5), x(t) is scalar elementary in its entire domain of existence (i.e. it has no points of violation of its scalar elementariness).

However the vice versa statement is an open Proposition.

Proposition. If a function x(t) is scalar elementary in its entire domain, it must satisfy some explicit polynomial ODE (5) rather than a rational ODE (3).

If a functions x(t) is scalar elementary in the entire domain of its existence, it means that the denominator g(t, x, x', x",...) in R=f/g of the rational ODE (3) must never disappear on x(t) in its entire domain. This seemingly suggests that there must be no denominator at all – as the image of complex functions usually is the entire complex space including 0 (unless the function is a constant), so that the Proposition seems true. In reality however...
• Either there is no denominator at all because the rational ODE (3) actually happened to be polynomial – as the Proposition claims;
• Or  g(t, x, x', x",...)   never reaches zero on x(t) because zero may happen to be the excluded (lacunary) value of g (in accordance with the Picard theorem) – see the Example below. It's because of exactly this possibility the Proposition remains an open statement.
Example. Observing that for the function x(t) = et the lacunar value is zero and x = x' = x", the following ODE takes place

x" = (x')2/x,     x(0)=x′(0)=1

and its denominator never disappears on x(t). Though in this trivial example we do know an alternative simple linear ODE satisfied by x(t) (in accordance with what the Proposition states), for an arbitrary x(t) we do not know whether the Proposition is true.

4) Regularization of a polynomial ODE and the Conjecture

First consider these examples.

Example 1. The (entire) function x(t) = tet  satisfies the ODE

tx' – tx – x = 0                                     (6)

singular at t=0. However x(t) satisfies also the ODE

x" – 2x' + x = 0

regular at t=0.

Example 2. The (entire) function x(t) = (et1)/t  satisfies the ODE

tx' – tx + x – 1 = 0                                 (7)

singular at t=0. However it's proven that there can not exist a polynomial ODE regular at t=0 satisfied by this function. Nor can it satisfy a polynomial ODE with a constant nonzero factor at the leading derivative – because unlike in the Example 1, here function x(t) does have a point of violation of its scalar elementariness.

Example 3. Besides the ODE (7), the function x(t) = (et1)/t  also satisfies the ODE

x"(x – 1)(t 1)(tx' – 2x' + x)(x' – 1)= 0

singular at t=1 (with any initial value for x, in particular with x(1) = e – 1 corresponding to x(t) in Example 2). At that, this x(t) satisfies also the ODE (7) regular at t=1 (yet still singular at t=0). If we set a goal to rid of singularity in at t=1 in the Example 3, ODE (7) is the answer (The ODE in Example 3 was obtained in a process of an intentional "planting" of a singularity into the ODE (7) at t=1 described here).

Now comes a question.

Let a holomorphic function x(t) satisfy a polynomial ODE over complex coefficients

P(t, x, x', ..., x(m)) = 0                                        (8)

which happened to be singular at t=0. Does a method exist which either offers another poly ODE

Q(t, x, x', ..., x(n)) = 0

satisfied by x(t) but regular at t=0, or it says that x(t) cannot satisfy any poly ODE (8) regular at t=0.

Here is how this question relates to the Conjecture.

If the Conjecture is true (and will be proved) then the answer to the question is this:

If a function x(t) at t=0 satisfies the polynomial system (2), then a regular replacement Q exists. And if function x(t) at t=0 cannot satisfy the polynomial system (2), then a regular replacement Q does not exist.

5) More suspects for points of violation of elementariness

More suspects for points of violation of elementariness appear in quite unexpected situations. For example, a transcendental function y(x) which is a non-trivial solution of the equation of commuting powers  xy=yx  also satisfies an ODE which is singular at the point x=y=e:

y"x2y2(y –  x) – (y')3x4 + (y')2x2y(3x –  2y) + y'xy2(3y –  2x) –  y4 = 0

and it is not known whether y(x) satisfies a regular ODE at the point x=y=e.

It is proven that y(x) is holomorphic at x=y=e, however no general term in closed form is known for its expansion (its derivatives at this point are obtainable only via a complicated recurrent equations) – see the article.

Does the y(x)  have violation of scalar elementariness at x=y=e

Being singular, at this point this ODE has more than one solution being satisfied also by the trivial solution – the identity function y=x . If it were proven that any ODE satisfied by the nontrivial solution y(x) is satisfied also by the trivial one (thus having more than one solution), that would prove that x=y=e is really the point of violation of the scalar elementariness of the nontrivial y(x). However this is not proved. Note, that the ODE y'=1 satisfied by the trivial solution is not satisfied by the nontrivial one.

Amazingly, a similar open question exists concerning the equation of the associating powers  x^(y^z) = (x^y)^z. It's solution y = z1/(z – 1)y(1) = e is proved to be holomorphic at the point  z = 1.  However it's not known if this is a point of violation of scalar elementariness.

If we denote z1=t  then y = z1/(z – 1)  =(1 + t)1/t = u(t), so that u(t) represents the remarkable limit:  lim u(t) = e  when  t→0. The proof that the function ln u(t) = (ln (1+t))/t loses scalar elementariness at t=0 is based on the special property and simplicity of its derivatives (–1)(n+1)n!/(n+1), however the derivatives of u(t) at t=0 are not so simple at all, the known proof does not work for them, while the fact that ln u(t) is scalar non-elementary does not tell anything whether u(t) is scalar non-elementary.

There is another unsolved problem associated with the commuting powers: the function y(x) happens to be closely approximated by a hyperbola (in red)

 h(x) = 1  + (e – 1)2 x – 1 and it was proved that h(x) y(x) in a small neighborhood of the point x=e. (The proof for all x>1 was recently obtained by Lajos Lóczi – August 2020).

6) "Automatic Differentiation" for singular epxressions

or
scenarios of evaluation of formal derivatives at a singular point of a polynomial m-order ODE.

As we saw above, a function f(t) holomorphic at some point t=t0 however may be defined with an expression having a removable singularity at this point so that f(t0) must be specially defined as the lim f(t) when  tt0 . Moreover, at certain points (where elementariness is lost) such functions can be defined only via singular rational expressions or ODEs.

Thus, we have holomorphic functions at a point, having all their derivatives, but the conventional AD is inapplicable for computing those derivatives. Therefore we developed a process (as a generalization of AD) capable to compute n-order derivatives of singular expressions or singular ODEs. Understandably, the AD generalized for singularities, is more complicate and less suitable for automatization. That is why the expression "Automatic Differentiation" is taken in quotes in the title. Unlike at a regular point, evaluation of ODEs at the singular points admits a variety of scenarios, which I first outlined in 2009, here in section 8.2.

All cases when an elementary function in question
x(t) is defined via elementary systems of ODEs or expressions containing singularities may be reduced to one implicit polynomial ODE  P(x(m) …, x', x, t) = 0  singular at some point t = a, (i.e. P/Xm = 0 at t = a). The solution x(t) may or may not exist at this point, or may be not unique. At a singular point we speak about a process of formal evaluation of the derivatives, and this process may have different outcomes outlined below.

The formal evaluation means applying the rules of differentiation to the source ODE not even expecting that the solution exists and has complex derivatives.

This process creates an infinite generating system for obtaining a vector of coefficients of the formal derivatives
x(m+1), x(m+2),…. Both in regular and singular cases the generating system is a triangular system.

In the regular case the generating system is linear, delivering a unique solution-vector for the formal
derivatives representing the unique holomorphic solution x(t) and the vector of its true derivatives (x, x', x",…, x(m), x(m+1),…).

In the singular case the scenario becomes much more complicated as displayed in the table below. If the process for evaluation of formal derivatives does not stop, the result of the generation is a finite or infinite set of solution vectors S ={(x, x', x",…, x(m), x(m+1) ,…)} comprised of formal derivatives allowing to write down the respective formal
Taylor expansions (having zero or non-zero convergence radius). This set S has a sophisticated tree-like structure showed in the following Table 2.

 At a regular point... At a singular point... The unique holomorphic solution x(t) exists, and its Taylor coefficients comprise the unique solution-vector of the generating system. A holomorphic solution of the ODE does not necessarily exist. The generating system has a unique solution-vector representing the derivatives x(n) of the solution x(t). The generating system may have no solution-vectors, or have more than one, or infinitely many of them. The generating system is triangular whose each equation is linear in one unknown with the same nonzero coefficient at every unknown. The generating system is triangular but not necessarily linear as each of its equations in one unknown is polynomial with numeric or parametric coefficients. Some of the equations may degenerate into a numeric relation true or false. If n-th numeric relation is false, the respective branch of the tree ends (marked by stop). If n-th numeric relation is true (i.e. a zero identity), the respective unknown is a free parameter (like c(m+2)). If one or several unknowns in the previous equations are free parameters, the subsequent equations become nonlinear polynomial equations with parametric (rather than numeric) coefficients, so that generally it gets impossible to numerically explore and evaluate them. If the set of the solution-vectors of the generating system is non-empty, some or all of the vectors still may comprise a divergent Taylor series (with zero radius) representing no solution.

Table 1

This table below illustrates an example of evaluation of a polynomial ODE at a singular point.

 The values of the formal derivatives obtainable from the equations of the generating system x x, x', x", .... , x(m)   are given. The equation in one unknown x(m+1)  happened to be of degree 3 having 3 roots  a1(m+1), a2(m+1), a3(m+1) . x' x" … x(m) x(m+1) a1(m+1) The equation in one unknown x(m+2)  happened to be of degree 2 having 2 roots  a1(m+2), a2(m+2) a2(m+1) The equation in one unknown x(m+2)  degenerated into an identity so that x(m+2)  may be an arbitrary complex number c(m+2) a3(m+1) x(m+2) a1(m+2) The equation in one unknown x(m+3)  degenerated into an ineqaulity a2(m+2) The equation in one unknown x(m+3)  happened to be of degree 3 c(m+2) The equation in one unknown x(m+3) and all similar equations further down contain an arbitrary parameter c(m+2), so that their roots depend on this parameter too –  the case of uncountable infinite branching. a3(m+2) a4(m+2) x(m+3) stop a1(m+3) a2(m+3) a3(m+3) a4(m+3)(c(m+2)) a5(m+3)(c(m+2)) a6(m+3)(c(m+2)) a7(m+3) a8(m+3) a9(m+3) a10(m+3) ... … … … … … … … … … … … … …

Table 2

Here  x, x', … , x(m) are given as the initial values, while x(m+1), x(m+2),…  are to be obtained in the process of solution of algebraic equations in one unknown of some degree k 0. If k=0 the respective equation degenerated into a numeric relation true or false. If false, this branch ended with a controversy. If true, this equation does not yield a specific solution value allowing the respective x(n) be an arbitrary constant (like c(m+2)). Therefore, the set of branches growing from every node of the tree is either…

• empty (end of the branch), or…
• an infinite uncountable set of the power continuum, or…
• a finite set with the number of branches k representing k different roots of the respective algebraic equation.

I have illustrative examples for many (though not all) cases of this evaluation scenario.

Where (in which textbook or monograph) is this setting and scenario considered?