Xu's Conjecture: Algebra, Graph Theory & MZVs

• Xu’s Conjecture is a term for unrelated conjectures in mathematics, spanning Jacobian pairs, normal Cayley graphs, and alternating double zeta values.
• In algebraic geometry, it conjectures an equality for intersection numbers using minor and major roots, though only an inequality is proven, impacting the Jacobian Conjecture program.
• In graph theory, it links the asymptotic prevalence of normal Cayley graphs to the Babai–Godsil GRR conjecture, while in multiple zeta values it yields a depth-preserving explicit identity.

Searching arXiv for papers explicitly referring to “Xu’s Conjecture” across different mathematical contexts. arxiv_search(query="^^^^^^^^2\2 Conjecture^^^^^^^^2\2 OR ^^^^^^^^2\2 conjecture^^^^^^^^2\2 max_results=^^^^2 OR \2\2^^^^, sort_by="relevance") Searching arXiv for the Jacobian-pair usage of Xu’s Conjecture. arxiv_search(query="(&&&^^^^2\2^^^^&&&) Jacobian Xu conjecture approximate roots intersection numbers", max_results=5, sort_by="relevance") Searching arXiv for the Cayley-graph usage of Xu’s Conjecture. arxiv_search(query="(&&&^^^^2 OR \2^^^^&&&) Xu conjecture Cayley graphs normal Cayley graphs", max_results=5, sort_by="relevance") Searching arXiv for the multiple-zeta-value usage of Xu’s Conjecture. arxiv_search(query="(Charlton, 4 Aug 2025) Xu conjecture alternating double zeta values", max_results=5, sort_by="relevance") Searching arXiv for more recent work on the Cayley-graph version to avoid relying only on a 2^^^^2\2 OR \2^^^^9 status statement. arxiv_search(query="Babai Godsil GRR conjecture normal Cayley graphs Xu 2^^^^2\22\2^^^^ 2^^^^2\22 OR \2^^^^ 2^^^^2\222^^^^ 2^^^^2\22^^^^3 2^^^^2\22^^^^4 2^^^^2\22^^^^5 2^^^^2\22^^^^6", max_results=^^^^2 OR \2\2^^^^, sort_by="submittedDate") Xu’s Conjecture is a label attached to several distinct conjectural statements associated with Xu. In the literature represented by the Jacobian-pair study (&&&^^^^2\2^^^^&&&), the Cayley-graph enumeration study (&&&^^^^2 OR \2^^^^&&&), and the alternating-double-zeta-value study (Charlton, 4 Aug 2025), the name refers to three unrelated problems. One concerns intersection numbers for Jacobian pairs, one concerns the asymptotic prevalence of normal Cayley graphs, and one concerns a depth-preserving descent of alternating double zeta values. This suggests that the designation is best interpreted contextually rather than as a single canonical conjecture.

^^^^2 OR \2^^^^. Distinct uses of the name

The three uses represented in these arXiv sources may be organized as follows.

Context	Conjectural statement	Status in cited source
Jacobian pairs	PRESERVED_PLACEHOLDER_^^^^2\2^^^^ for the minor-roots expression	translated only as PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^ (&&&^^^^2\2^^^^&&&)
Cayley graphs	$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$ as $r\to\infty$	proved equivalent to Babai–Godsil’s GRR conjecture (&&&^^^^2 OR \2^^^^&&&)
Alternating double zeta values	$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$ is expressible in terms of non-alternating double zeta values	proved with an explicit formula (Charlton, 4 Aug 2025)

The ambiguity is substantive rather than terminological. In (&&&^^^^2\2^^^^&&&), Xu’s Conjecture is embedded in the two-variable Jacobian Conjecture and concerns the exact computation of an affine intersection number from minor root data. In (&&&^^^^2 OR \2^^^^&&&), Xu’s Conjecture is an asymptotic statement in algebraic graph theory about normal Cayley graphs. In (Charlton, 4 Aug 2025), Xu’s Conjecture is a concrete family of identities in the algebra of alternating multiple zeta values.

2. Xu’s Conjecture in the Jacobian-pair setting

In the Jacobian-pair setting, the ambient problem is Keller’s two-variable Jacobian Conjecture. If $P,Q\in L:=K[x,y]$ satisfy

$$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$

then the Jacobian Conjecture in dimension two asserts that

$(x,y)\longmapsto (P(x,y),\,Q(x,y))$

is an automorphism of $L$. A pair $(P,Q)$ with PRESERVED_PLACEHOLDER_^^^^2 OR \2\2^^^^ is a Jacobian pair (&&&^^^^2\2^^^^&&&).

The paper works in the framework of Valqui–Guccione–Guccione. For a normalized potential counterexample, one studies the affine intersection number

PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^

Xu’s program, as translated in (&&&^^^^2\2^^^^&&&), uses Newton–Puiseux expansions and approximate roots, called PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^-roots, to partition the roots of PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^3 into clusters attached to final PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^4-roots. These final PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^5-roots are divided into major and minor classes according to

PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^6

with PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^7 for major final PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^8-roots and PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^9 for minor final $\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$^^^^2\2^^^^-roots.

The exact major-roots formula obtained in (&&&^^^^2\2^^^^&&&) is

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$^^^^2 OR \2^^^^

where $\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$2. This matches Xu’s “major roots” intersection formula. The conjectural point concerns the minor roots. The paper defines

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$3

with

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$4

for minor $\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$5. In the language of (&&&^^^^2\2^^^^&&&), Xu’s Conjecture is the assertion that the minor-roots formula is an equality, so that the exact intersection number can also be written as

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$6

3. Translation, inequality, and the failure of equality in the Jacobian formulation

The central conclusion of (&&&^^^^2\2^^^^&&&) is that the major-roots identity survives translation, whereas the minor-roots identity does not. Theorem 3.25 gives

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$7

together with

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$8

and

$\min_{|R|=r}\frac{|N(R)|}{|T(R)|}\to 1$9

Using

$r\to\infty$^^^^2\2^^^^

the paper obtains only the bound

$r\to\infty$^^^^2 OR \2^^^^

rather than equality (&&&^^^^2\2^^^^&&&).

The obstruction appears in the estimate

$r\to\infty$2

for $r\to\infty$3. Equality is not guaranteed, and Lemma 3.23(6) explicitly allows

$r\to\infty$4

Accordingly, the contribution of minor root clusters is bounded above by $r\to\infty$5, but may be strictly smaller. The paper does not give an explicit counterexample where the inequality is strict, but it shows that Xu’s argument does not justify the equality he needs. The authors therefore confirm the major-root side of Xu’s program, but replace the minor-root identity by an inequality (&&&^^^^2\2^^^^&&&).

This has direct consequences for the Jacobian Conjecture program. The introduction states that if both formulas were true, one would be able to discard many infinite families of possible counterexamples described in [8]. After translation into the language of [5], the same formula is obtained for $r\to\infty$6, but for $r\to\infty$7 only an inequality is obtained; consequently the infinite families cannot be discarded as desired (&&&^^^^2\2^^^^&&&).

4. Xu’s Conjecture in Cayley-graph enumeration

In algebraic graph theory, Xu’s Conjecture is an asymptotic statement about normal Cayley graphs. For a finite group $r\to\infty$8 and $r\to\infty$9, the Cayley digraph $$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$^^^^2\2^^^^ has vertex set $$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$^^^^2 OR \2^^^^ and arc set determined by

$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$2

When $$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$3, the digraph is undirected and is called a Cayley graph. A Cayley graph is normal if

$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$4

A graphical regular representation (GRR) is a Cayley graph satisfying

$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$5

(&&&^^^^2 OR \2^^^^&&&).

The relevant counting sets are

$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$6

and

$$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$7

Spiga restates Xu’s conjecture in the undirected setting as follows: $$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$8 Equivalently, for every $$\zeta(2,\overline{2k}) + 2k\,\zeta(1,\overline{2k+1})$$9 there exists $P,Q\in L:=K[x,y]$^^^^2\2^^^^ such that for all groups $P,Q\in L:=K[x,y]$^^^^2 OR \2^^^^ with $P,Q\in L:=K[x,y]$2,

$P,Q\in L:=K[x,y]$3

In words, for every large group $P,Q\in L:=K[x,y]$4, almost all Cayley graphs on $P,Q\in L:=K[x,y]$5 are normal, uniformly over all groups of a given order (&&&^^^^2 OR \2^^^^&&&).

The paper emphasizes that the sentence “normal Cayley digraph” in the statement is a typographical slip: because the count is over inverse-closed subsets $P,Q\in L:=K[x,y]$6, the intended objects are undirected Cayley graphs. Xu also formulated an analogous conjecture for Cayley digraphs without the inverse-closed restriction, and that directed version was already proved via Babai–Godsil’s DRR conjecture (&&&^^^^2 OR \2^^^^&&&).

Two structural obstruction families are singled out throughout the discussion: abelian groups of exponent greater than $P,Q\in L:=K[x,y]$7, and generalised dicyclic groups. These are exactly the infinite families that never admit GRRs (&&&^^^^2 OR \2^^^^&&&).

5. Equivalence with the Babai–Godsil GRR conjecture

The main theorem of (&&&^^^^2 OR \2^^^^&&&) is: $P,Q\in L:=K[x,y]$8 Here Conjecture ^^^^2 OR \2^^^^.2 is Babai–Godsil’s GRR conjecture for finite groups $P,Q\in L:=K[x,y]$9 that are neither generalized dicyclic nor abelian of exponent greater than $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$^^^^2\2^^^^, and Conjecture ^^^^2 OR \2^^^^.4 is Xu’s conjecture on the asymptotic prevalence of normal Cayley graphs. Thus GRR-genericity and normality-genericity are asymptotically equivalent (&&&^^^^2 OR \2^^^^&&&).

The proof isolates one principal obstruction: $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$^^^^2 OR \2^^^^ that is, the existence of nonidentity automorphisms of $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$2 preserving the connection set $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$3. Lemma 2.7 gives a dichotomy for a nonidentity automorphism $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$4: either the number of $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$5-invariant inverse-closed subsets is at most

$$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$6

for some absolute constant $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$7, or one is in one of the two obstruction families. Summing over all nontrivial automorphisms yields Proposition 2.8: $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$8 for groups $$[P,Q] := \partial_x P\,\partial_y Q - \partial_x Q\,\partial_y P \in K^\times,$$9 outside the two exceptional families (&&&^^^^2 OR \2^^^^&&&).

Taking ratios with the total number

$(x,y)\longmapsto (P(x,y),\,Q(x,y))$^^^^2\2^^^^

of inverse-closed subsets gives Theorem ^^^^2 OR \2^^^^.3: the proportion of inverse-closed $(x,y)\longmapsto (P(x,y),\,Q(x,y))$^^^^2 OR \2^^^^ such that

$(x,y)\longmapsto (P(x,y),\,Q(x,y))$2

goes to $(x,y)\longmapsto (P(x,y),\,Q(x,y))$3 as $(x,y)\longmapsto (P(x,y),\,Q(x,y))$4, for groups $(x,y)\longmapsto (P(x,y),\,Q(x,y))$5 that are neither generalized dicyclic nor abelian of exponent greater than $(x,y)\longmapsto (P(x,y),\,Q(x,y))$6. This asymptotic elimination of group-automorphism obstructions is the key input in the equivalence proof (&&&^^^^2 OR \2^^^^&&&).

Conceptually, the result shows that resolving Xu’s conjecture for undirected Cayley graphs is equivalent to resolving the stronger Babai–Godsil conjecture once the first obstruction has been removed asymptotically. The remaining difficulty lies in normal Cayley graphs whose extra automorphisms are not induced by automorphisms of the underlying group (&&&^^^^2 OR \2^^^^&&&).

6. Xu’s Conjecture for alternating double zeta values

In the multiple-zeta-value setting, Xu’s Conjecture concerns alternating depth-$(x,y)\longmapsto (P(x,y),\,Q(x,y))$7 Euler sums. For integers $(x,y)\longmapsto (P(x,y),\,Q(x,y))$8 and signs $(x,y)\longmapsto (P(x,y),\,Q(x,y))$9, the alternating multiple zeta values are

$L$^^^^2\2^^^^

with $L$^^^^2 OR \2^^^^. A bar on an index indicates a minus sign in the corresponding summand, so

$L$2

(Charlton, 4 Aug 2025).

Xu’s Conjecture, stated as Conjecture 7.2 in Xu’s weight $L$3 study, is that for positive integers $L$4, the combination

$L$5

can be expressed in terms of non-alternating double zeta values. The 2^^^^2\22^^^^5 paper proves this conjecture explicitly. Its main theorem states that for any integer $L$6,

$L$7

This expresses the left-hand side entirely in terms of depth-$L$8 non-alternating double zetas, products $L$9, and $(P,Q)$^^^^2\2^^^^. In particular, the expression lies in the $(P,Q)$^^^^2 OR \2^^^^-span of non-alternating MZVs of depth $(P,Q)$2 (Charlton, 4 Aug 2025).

The proof combines three ingredients. First, the target combination is rewritten as a shuffle-regularised alternating double zeta value: $(P,Q)$3 Second, a dihedral symmetry relation relates the regularised value to $(P,Q)$4 and depth-$(P,Q)$5 alternating zetas. Third, an explicit Galois descent formula expresses $(P,Q)$6 in terms of ordinary double zetas and products of single zetas (Charlton, 4 Aug 2025).

A central theme of the paper is the limitation of motivic Galois descent with respect to depth. Glanois’ criterion can show that a motivic alternating combination lies in the motivic subspace generated by non-alternating MZVs, but it does not control the depth of the resulting expression. The paper gives the example of $(P,Q)$7, whose explicit level-$(P,Q)$8 descent involves the depth-$(P,Q)$9 non-alternating MZV PRESERVED_PLACEHOLDER_^^^^2 OR \2\2\2^^^^. Xu’s Conjecture is therefore stronger than a mere level-PRESERVED_PLACEHOLDER_^^^^2 OR \2\2 OR \2^^^^ to level-PRESERVED_PLACEHOLDER_^^^^2 OR \2\22^^^^ descent: it is a depth-preserving descent statement, and the 2^^^^2\22^^^^5 result proves it by an explicit formula rather than by motivic descent alone (Charlton, 4 Aug 2025).

7. Comparative assessment

Taken together, these sources show that “Xu’s Conjecture” names three mathematically unrelated conjectural mechanisms. In the Jacobian-pair setting, the conjectural equality

PRESERVED_PLACEHOLDER_^^^^2 OR \2\2^^^^3

is not recovered by the careful translation of (&&&^^^^2\2^^^^&&&); only

PRESERVED_PLACEHOLDER_^^^^2 OR \2\2^^^^4

is proved. In the Cayley-graph setting, Xu’s conjecture is not a local identity but a uniform asymptotic statement, and (&&&^^^^2 OR \2^^^^&&&) shows that it is equivalent to Babai–Godsil’s GRR conjecture for undirected Cayley graphs. In the alternating-double-zeta-value setting, Xu’s conjecture is a concrete infinite family of identities, and (Charlton, 4 Aug 2025) proves it with an explicit formula.

The contrast in status is sharp. One version is weakened from equality to inequality, one is reformulated as an equivalence with another major asymptotic conjecture, and one is fully established. A plausible implication is that the name “Xu’s Conjecture” is best treated as a context-dependent label whose meaning is fixed entirely by subject area: affine algebraic geometry, algebraic graph theory, or the theory of multiple zeta values.