Are (integers) plus (elementary functions) plus (generalized hypergeometric functions) sufficient to represent any algebraic number?

For example, the real algebraic number $\alpha\in(-1,0)$ satisfying $$65536\,\alpha^{10}+327680\,\alpha^9+327680\,\alpha^8-655360\,\alpha ^7-983040\,\alpha^6+16720896\,\alpha^5\\+20983040\,\alpha^4-655360\,\alpha ^3-109155805\,\alpha^2-30844195\,\alpha +16762589=0$$ can be represented as $$\alpha={_4F_3}\left(\begin{array}c\frac15,\frac25,\frac35,\frac45\\\frac12,\frac34,\frac54\end{array}\middle|\frac1{\sqrt5}\right)-\frac{1+\sqrt5}2.$$ (see Bring radical for details)

Here are answers where I used some particular cases when this representation is possible: [1], [2]. These cases are motivating to try to find a general method applicable to all algebraic numbers.

  • $\begingroup$ This is the same as asking whether all non-radicals, not just those of order $5$ can, can be expressed as hypergeometric functions. $\endgroup$ – Lucian Nov 27 '13 at 5:35
  • $\begingroup$ At first I thought this decic was special and had something to do with the golden ratio. But, as you point out, the root $\alpha$ is the root of a Bring quintic affixed to (any) quadratic irrational, hence the golden ratio was just a nice but arbitrary choice. $\endgroup$ – Tito Piezas III Sep 21 '16 at 2:13

I'll give you a yes and a no, depending on exactly what you are looking for:


See Sturmfels' paper "Solving Algebraic Equations in Terms of $A$-Hypergeometric Series"

Consider a root of $x_0 + x_1 t + \cdots + x_n t^n$ as a function $F(x_0, x_1, \ldots, x_n)$. Then $F$ is an A-hypergeometric function (also known as a GKZ-hypergeometric function), associated to the $A$-matrix $$\begin{pmatrix} 0 & 1 & 2 & \cdots & n \\ 1 & 1 & 1 & \cdots & 1 \end{pmatrix}.$$ This result was in some sense known before, but Sturmfels writes these functions down very explicitly.

See Cattani's lectures for a quick intro to A-hypergeometric functions, including the definitions of the terms used above. This fact is Remark 3.23, and Cattani gives the relevant history.

I must admit that I never learned how to translate the modern A-hypergeometric language into the classical Gauss vocabulary. Note, however, that A-hypergeometric functions are power series in many variables, not one. And that brings me to:


Abhyankar has a paper which I wish I understood better; I tried to explain my partial understanding here. Let $F(x_5, x_6)$ be a root of $t^6+x_5 t + x_6=0$. I believe that Abhyankar is showing that $F$ can not be expressed in terms of field operations and holomorphic functions of single variables; it intrinsically has to be a function of two variables. So as long as you stay with $F \left( \begin{smallmatrix} a_1 & a_2 & \cdots & a_k \\ b_1 & b_2 & \cdots & b_{\ell} \end{smallmatrix} \mid z\right)$ and connect this together with field operations, I think Abhyankar is telling you you can never express the roots of a general sextic.

Both of my comments address the question of expressing the roots of $x_n t^n + \cdots + x_1 t + x_0$ as functions of $(x_n, \ldots, x_1, x_0)$ by uniform formulas. I am not addressing the question of whether any particular algebraic number might happen to be equal to a hypergeometric function. I see that you already asked a very hard question along those lines about expressing algebraic numbers in terms of exponentials; I expect that proving anything about hypergeometric functions can only be harder.

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  • $\begingroup$ Very interesting. $\endgroup$ – El Ectric Mar 28 '19 at 0:12

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