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It is well-known that CAS are able to perform operations like formal differentiation (relatively easily) and formal integration (via the Risch algorithm) in an algorithmic way.

But is there anything equivalent for the resolution of transcendental equations, be it for the expression of the roots, or for root isolation ?

Of course root computation is known to be feasible symbolically for polynomials up to degree four and generally infeasible for higher degrees, and some equations can be transformed into polynomial form by variable substitution. For instance,

$$8^x-7\cdot2^x-6=0$$ has closed-form expression for the root(s).

But is there any theory for general equations, discussing the number of roots, approximations to the roots or the exact roots themselves ? For instance, it is an easy matter to prove that $\sin x=x$ has a single root at $x=0$ or that $\tan x=x$ has single roots in intervals of length $\pi$. But can a CAS infer this ?

Acceptable solutions could be purely symbolic, but also numerical, provided in the latter case guarantees are given that all roots are enumerated.

What is known about this topic ?

  • Another example is $\sum_k c_k\sin(kt)$ is a polynomial in complex exponentials. – mathreadler Apr 17 '17 at 10:17
  • @mathreadler: I am not just looking for equations that can be reduced to algebraic ones. I am asking about all the others. But a theory of equations reducible to algebraic would be interesting as well. –  Apr 17 '17 at 10:19
  • If the CAS can prove sin and cos are periodic then it can probably prove that the quotients must be. It can try do proofs on limits and smoothness, deriving that the range of tan must be all $R$. But then it needs to prove invertibility on each period to show the solution $=x$ is unique. If it can do differentiation then that should be enough since differentiation shows that function is strictly monotonic and therefore must be invertible. – mathreadler Apr 17 '17 at 10:29
  • @mathreadler To show that a function is monotonic, you must discuss the roots of its derivative. –  Apr 17 '17 at 10:39
  • Yes and they are $\frac 1 {1+x^2}$ which is always positive since real squares are positive, addition of positives are positive and quotient of positives are positive. No wait, that's arctan. Oops. – mathreadler Apr 17 '17 at 10:41
  • @mathreadler: is there a general theory ? –  Apr 17 '17 at 10:42
  • We can still use the same arguments though, closure of sets under operations. – mathreadler Apr 17 '17 at 10:43
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    One general theory is Nevanlinna theory. – Antonio Vargas Apr 17 '17 at 11:25
  • Some papers that might of interest: Chee Yap and Lihong Zhi as well as the references included therein. There are also some special cases where more can be said, e.g., D-finite functions. – Michael Burr May 30 '19 at 16:51

1 Answers1

1

Formal differentiation and formal integration mean symbolic operations and symbolic results or the decision that no symbolic result could be found. Both problems are comprehensively discussed in the literature, but symbolic solving of transcendental equations seemingly not. I don't know what the algorithms for symbolic solving of transcendental equations in computer algebra systems are. Maybe there are no generally accepted and used general algorithms. $\ $

I want to try to describe the procedure of solving an explicitly given equation of one unknown symbolically (means: in closed form). You are invited to help to complete and improve it.

1.) Solving an equation means to determine its solution set.

2.) An explicitly given equation can be transformed into its zeroing equation by substracting the right-hand side of the equation.

3.) Let's consider the zeroing equation

$$F(x)=0$$

wherein $F$ is a function in dependence of $x$.

4.) Operations and functions are synonymous. That means, all operations in and with this equation are functions.

5.) Calculate the factorization of $F(x)$ as polynomial with the transcendental functions as variables. Set the factors of $F(x)$ successively equal to $0$. This gives further equations of the unknown $x$. Solve this equations for $x$.

6.) $0$ is an algebraic number. Look for which algebraic $x$ $F(x)$ can be algebraic (the algebraic points of $F$). If $F$ doesn't have algebraic points, the equation doesn't have algebraic solutions. Often, one can find simple algebraic solutions in this way.

7.) An equation $F(x)=0$ can be solved for $x$ if the values of partial compositional inverses $F^{-1}$ of $F$ at $0$ are defined and known.

8.) An equation can be solved if it can be rearranged for $x$. An equation $F(x)=0$ can be rearranged for $x$ if a partial compositional inverse $F^{-1}$ of $F$ is known and defined at $0$.

9.)

$n\in \mathbb{N}_+$
$\forall i\in \{1,...,n\}\colon f_i\colon x\mapsto f_i(x)$
$F\colon x\mapsto F(x)=f_1(f_2(...(f_{n-1}(f_n(x)))...)))$

An equation $F(x)=0$ be given. If there exist $f_1,...,f_n$ which are unary univalued functions, the equation can be solved by applying the partial compositional inverses $f_1^{-1},...,f_n^{-1}$ of $f_1,...,f_n$: $$x=f_n^{-1}(f_{n-1}^{-1}(...(f_2^{-1}(f_1^{-1}(0)))...))).$$

If $f_1,...,f_n$ and $f_1^{-1},...,f_n^{-1}$ are symbolically given, the equation can be solved symbolically by rearranging it for $x$ by applying only the partial inverses $f_1^{-1},...,f_n^{-1}$.

10.)

In [Ritt 1925] and [Risch 1979], the following is proved:
If $F$ is an elementary invertible elementary function over an open domain, there must exist such composition form with only unary elementary functions $f_1,...,f_n$.

That means, an elementary function over an open domain for which no such form exists doesn't have an elementary inverse.

For non-elementary functions or non-elementary inverses, this seems to be still an open problem:
https://mathoverflow.net/questions/320801/how-to-extend-ritts-theorem-on-elementary-invertible-bijective-elementary-funct

(The problem of integrability of elementary functions by elementary functions and the problem of invertibility of elementary functions by elementary functions
- use the same presentation of the elementary functions as decomposition into a generalized composition of algebraic functions, $\exp$ and/or $\ln$,
- can be treated in the language of differential algebra with (elementary) field extensions,
- can be related to Risch's structure theorem about the algebraic independence of elementary functions,
- are systematic and complete solutions of the problem,
- could be generalized to further classes of functions.)

11.) Even if $F$ doesn't have a partial inverse in closed form, closed-form solutions of the equation could exist. This problem is asked in [Chow 1999]. [Lin 1983] and [Chow 1999] prove that the polynomials $P\in\overline{\mathbb{Q}}[x,e^x]$ cannot have solutions that are elementary numbers and explicit elementary numbers respectively. For further classes of elementary functions, this seems to be still an open problem: Which kinds of equations of elementary functions can have elementary solutions?

$\ $

Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer, 2014 $\ \ \ \ \ \ $ mainly means differential equations

[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50

[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

Wikipedia: Elementary function

IV_
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  • I am afraid that this works for few cases. $(x+1)\log x$ has the obvious root $x=1$ (closed-form), though the function has no analytical inverse, nor does its derivative. –  May 25 '19 at 15:55
  • @ Yves Daoust Could you please delete your comment? It is no more needed. I added the known usual solution procedue you mentioned to my answer. I think we must not discuss so much on MathStackexchange. – IV_ May 26 '19 at 15:05