Resolvent Degree of the General Degree-Seven Polynomial (Algebraic Form of Hilbert's Thirteenth Problem)

Resolvent Degree of the General Degree-Seven Polynomial (Algebraic Form of Hilbert's Thirteenth Problem)

Definitions and mathematical statement

Resolvent degree over ${\mathbb{C}}$ measures how many independent parameters are needed in the algebraic functions used to solve a generic equation. Formally, it is the least $d$ for which the generic root cover is dominated by a finite tower of covers whose steps arise by rational pullback from algebraic covers of varieties of dimension at most $d$, using the standard birational convention of the source. For the general polynomial $X^7+a_1X^6+\cdots+a_7$ with algebraically independent coefficients, the target is \[ \operatorname{rd}_{{\mathbb{C}}}(7)=3, \] or, using the upper bound incorporated in the catalogue, the obstruction $\operatorname{rd}_{{\mathbb{C}}}(7)>2$. This is an algebraic-function problem, not the continuous-superposition version of Hilbert's thirteenth problem.

Formulation and concept references: [S1].

Short English statement

Does solving the generic degree-seven polynomial necessarily require algebraic functions of at least three independent variables, even when finite towers of such functions are allowed?

Research attempt: what monodromy does and does not obstruct

GPT-6 Astra Ultra, 27 September 2026. Status: UNRESOLVED.

The required lower bound is stronger than nonsolvability by radicals and different from a parameter count for one preferred normal form. Towers of algebraic functions are allowed, and the parameters at a later step may be rational functions of quantities produced earlier. I verify the generic Galois obstruction, construct a one-parameter cover with the same large monodromy, and isolate why neither gives the requested lower bound of three.

The generic polynomial has Galois group $S_7$.

Let $x_1,\ldots,x_7$ be independent transcendentals and put $e_i=e_i(x_1,\ldots,x_7)$. The elementary symmetric functions are algebraically independent, and the field of symmetric rational functions is \[ \mathbb C(x_1,\ldots,x_7)^{S_7} =\mathbb C(e_1,\ldots,e_7). \] The polynomial $\prod_i(X-x_i)$ is the generic monic degree-seven polynomial after replacing coefficients by signed $e_i$. The permutation action is faithful, so its splitting extension is Galois with group $S_7$.

Since $S_7$ is not solvable, the generic polynomial is not solvable by a tower of radical extensions. Over a characteristic-zero field containing roots of unity, the Galois closure of a radical tower has solvable Galois group; the generic root cover cannot be dominated by such a tower. This is a valid obstruction, but a one-variable algebraic function need not be a radical. The definition in the question permits arbitrary finite algebraic covers of curves, not just covers $z^m=t$.

The failure is not a small technicality. It can be demonstrated with another degree-seven polynomial whose monodromy is exactly $S_7$ and whose root cover manifestly uses only one parameter.

A one-parameter $S_7$ cover.

Consider the map \[ p:\mathbb P^1_z\longrightarrow\mathbb P^1_t, \qquad t=z^7-z. \tag{1} \] Its six finite critical points satisfy $7z^6=1$. They are simple because $p''(z)=42z^5\ne0$ there. At such a point $r$, \[ p(r)=r/7-r=-6r/7, \] so the six critical values are distinct. A small loop around a finite branch value exchanges the two local sheets meeting at its simple ramification point and fixes the others; its monodromy is a transposition.

The cover is connected, so the monodromy group is transitive on the seven sheets. The finite branch loops generate its monodromy: the loop around infinity is the inverse of their product. Construct a graph with the seven sheets as vertices and one edge for each transposition. The group generated by edge transpositions acts independently on the graph components, and within a connected component it generates the full symmetric group. Transitivity forces this graph to be connected. Therefore the monodromy of $1$ is $S_7$.

Yet $1$ is a cover of a one-dimensional base. Its own resolvent degree is at most one, simply by taking the cover itself as an allowed step. Thus neither nonsolvability, the presence of $A_7$, nor the full monodromy group $S_7$ by itself forces resolvent degree greater than two for an individual cover.

There is no contradiction with the target. The cover $1$ is one particular one-parameter family. It has not been shown to be a universal family from which the generic seven-coefficient root problem can be obtained through a tower of low-dimensional pullbacks. The distinction between a group realized as monodromy somewhere and the complexity of its generic torsors is essential. The formal framework in Farb--Wolfson [E1] is designed to keep these questions separate.

A direct normalization gives an upper bound, not a lower bound.

For the generic polynomial \[ X^7+a_1X^6+\cdots+a_7, \] make the rational translation $X=Y-a_1/7$. The coefficient of $Y^6$ vanishes, leaving \[ Y^7+b_2Y^5+b_3Y^4+b_4Y^3+b_5Y^2+b_6Y+b_7. \] On the dense open set $b_7\ne0$, adjoin $\lambda$ with $\lambda^7=b_7$. This is an allowed one-parameter radical cover. Substituting $Y=\lambda Z$ and dividing by $\lambda^7$ yields \[ Z^7+c_2Z^5+c_3Z^4+c_4Z^3+c_5Z^2+c_6Z+1, \qquad c_j=b_j\lambda^{-j}. \tag{2} \] The family in $2$ uses five parameters. Solving its root cover and then undoing the transformations gives the elementary bound $\operatorname{rd}_{\mathbb C}(7)\le5$.

This is weaker than the classical upper bound three incorporated in the catalogue and discussed in [E1]. Its purpose is to expose exactly which operations justify a parameter reduction. The translation is rational; the scaling is a permitted algebraic step; and the reconstruction of $X$ from $Z$ is explicit. Merely asserting that two coefficients can be "normalized away" without accounting for the scaling cover would omit part of the allowed tower.

Nor does the persistence of five coefficients in this particular normal form prove a lower bound of five, three, or even two. A different transformation or a sequence of covers could compress the problem further. To prove the requested obstruction one must exclude every tower of the specified type, not just transformations of degree one in the root variable.

A tower defeats naive counts of independent inputs.

Let $t_1,\ldots,t_r$ be independent transcendentals. Over $K=\mathbb C(t_1,\ldots,t_r)$, adjoin \[ \sqrt{t_1},\ \sqrt{t_2},\ \ldots,\ \sqrt{t_r} \] one at a time. Every step is a pullback of the same one-variable cover $z^2=t$. The total extension has Galois group $(\mathbb Z/2\mathbb Z)^r$ and degree $2^r$: independence of the square classes follows, for example, by taking valuations at the divisors $t_i=0$.

Thus arbitrarily many independent base parameters and arbitrarily large extension degree can coexist with resolvent degree at most one. The base still has transcendence degree $r$ after every finite extension; that transcendence degree is not the dimension of the small variety from which an individual step is pulled back. A proof that subtracts the number of radical adjunctions from the number of base parameters would confuse these distinct invariants.

This example also clarifies adaptivity. A later parameter may be a rational function on a variety already obtained in the tower, so it can depend algebraically on earlier roots. Ruling out a single simultaneous two-parameter formula is weaker than ruling out a finite tower of two-parameter functions. Continuous superposition results are even further removed: their building blocks need not define algebraic covers at all.

The elementary lower bound that does survive.

Over the algebraically closed constant field $\mathbb C$, finite covers of zero-dimensional varieties have no nontrivial function-field extension. Pulling them back can only add split components; a tower of such steps cannot dominate a connected nontrivial generic root cover. Hence \[ \operatorname{rd}_{\mathbb C}(7)\ge1. \] Together with the established upper bound quoted in the statement, this leaves $1\le\operatorname{rd}_{\mathbb C}(7)\le3$. The arguments above do not exclude one or two. The nonsolvable Galois group excludes radical towers, which form only a proper subclass of the allowed one-parameter towers.

A possible successful lower-bound invariant would need two properties at once: a nonzero value on the generic degree-seven cover, and a vanishing or monotonicity rule stable under all towers of pullbacks from surfaces. The computations here provide no such invariant. A monodromy invariant depending only on the abstract group fails the individual-cover test $1$, while a raw parameter count fails the independent-square-root test.

Verification and outcome.

The local exact checks expand the translation leading to $2$, verify the scaling exponents, and generate $S_7$ from a connected set of transpositions on seven letters. They also check the branch-value relation $p(r)=-6r/7$ modulo $7r^6-1$. The monodromy proof uses topology of the stated cover; enumerating permutations is a consistency check, not a new lower-bound theorem.

The result is a precise separation of radical complexity, monodromy, and resolvent degree, together with explicit normalizations and countertests to two natural lower-bound routes. The first unresolved step remains an obstruction stable under arbitrary finite towers of covers of dimension at most two. No such obstruction is established here, so the algebraic degree-seven problem is unresolved by this attempt.

Sources

• S1 A. G. Vitushkin, "On Hilbert's thirteenth problem and related questions", Russian Mathematical Surveys 59 $2004$, no. 1; DOI 10.1070/RM2004v059n01ABEH000698. https://iopscience.iop.org/article/10.1070/RM2004v059n01ABEH000698. Formulation source.
• E1 B. Farb and J. Wolfson, Resolvent degree, Hilbert's 13th Problem and geometry. Used for the tower definition, distinction between group and individual-cover complexity, and the established upper bounds. https://arxiv.org/abs/1803.04063.