Pellarin's operator-theoretic Schanuel conjecture for the Carlitz exponential over Tate algebras: regular $\mu$-independence of $\exp_C$-values

In function-field arithmetic, the Carlitz exponential $\exp_C$ plays the role of the classical exponential, and Denis (1995) proposed a Schanuel-type conjecture for it over the completed algebraic closure $\mathbb{C}_\infty$ of $\mathbb{F}_q((1/\theta))$. Pellarin (2017) extended $\exp_C$ to complete difference fields $\mathbb{K}_s$ built from Tate algebras in variables $t_1,\dots,t_s$, and proposed a stronger, operator-theoretic Schanuel-type conjecture (Conjecture 18 of the arXiv version, Conjecture 3.14 of the published version): whenever $f_1,\dots,f_n\in\mathbb{K}_s$ are $\mathbb{F}_q(t_1,\dots,t_s)[\theta]$-linearly independent, at least $n$ of the $2n$ elements $f_1,\dots,f_n,\exp_C(f_1),\dots,\exp_C(f_n)$ are regularly $\mu$-independent over an explicitly defined countable base field $L_s$, where regular $\mu$-independence means non-vanishing of all analytically regular $\mu$-difference polynomials. The paper proves implications but no new case beyond those already known for Denis' conjecture, and the conjecture — already open for $s=0$, where it reduces to Denis' conjecture — has neither been proved nor refuted in the subsequent literature.

Background

Let $\mathbb{F}_q$ be the finite field with $q=p^e$ elements, $A=\mathbb{F}_q[\theta]$, and $K=\mathbb{F}_q(\theta)$. Completing $K$ at its infinite place, then taking an algebraic closure and completing again, yields the field $\mathbb{C}_\infty$, the function-field analogue of $\mathbb{C}$. The Carlitz module endows any $A$-algebra $B$ with an $A$-module structure in which multiplication by $\theta$ acts through the Frobenius twisting operator $\tau(b)=b^q$ via $C_\theta(b)=\theta b+b^q$. The Carlitz exponential is the entire, $\mathbb{F}_q$-linear function $\exp_C(z)=\sum_{i\ge0}d_i^{-1}z^{q^i}$, where $d_i$ is the product of the monic elements of $A$ of degree $i$ (an analogue of $q^i!$); it uniformizes the Carlitz module, and its kernel is the rank-one $A$-module $\widetilde{\pi}A$, where the Carlitz period $\widetilde{\pi}$ is transcendental over $K$ (Wade, 1941). Denis (1995) conjectured the direct Schanuel analogue: if $u_1,\dots,u_n\in\mathbb{C}_\infty$ are $A$-linearly independent, then $K(u_1,\dots,u_n,\exp_C(u_1),\dots,\exp_C(u_n))$ has transcendence degree at least $n$ over $K$. This remains open; the known cases are the Hermite–Lindemann analogue for the Carlitz exponential (Wade, and in general a theorem of Yu on Drinfeld modules, the higher-rank generalizations of the Carlitz module), the Lindemann–Weierstrass analogue for $u_i$ algebraic over $K$ (Thiery), and the case where all $\exp_C(u_i)$ are algebraic (Papanikolas, via Tannakian duality for Anderson $t$-motives — certain finite-rank difference modules that play the role of motives in this setting).

Pellarin's paper moves the problem into a several-variable rigid-analytic setting. For $s\ge0$, the Tate algebra $\mathbb{T}_s$ is the completion of $\mathbb{C}_\infty[t_1,\dots,t_s]$ for the Gauss norm (the norm given by the maximum of the norms of the coefficients), and $\mathbb{K}_s$ is the complete field of generalized formal series $\sum_i c_i\theta^{-i}$ ($c_i\in\mathbb{F}_q^{ac}(t_1,\dots,t_s)$, well-ordered rational support), containing $\mathbb{C}_\infty(t_1,\dots,t_s)$. The map $\mu:x\mapsto x^p$ extends to a continuous, $\mathbb{F}_p(t_1,\dots,t_s)$-linear automorphism of $\mathbb{T}_s$ and $\mathbb{K}_s$, making them difference fields; since $\tau=\mu^e$, the Carlitz exponential extends by $\exp_C=\sum_{i\ge0}d_i^{-1}\mu^{ei}$ to a map $\mathbb{K}_s\to\mathbb{K}_s$ with kernel $\widetilde{\pi}\,\mathbb{F}_q(t_1,\dots,t_s)[\theta]$ (Anglès–Pellarin–Tavares Ribeiro). The base field is $K_s=\mathbb{F}_p(\theta,t_1,\dots,t_s)$.

On such difference fields, algebraic relations are encoded by $\mu$-polynomials: elements of the ring $L[\underline{X}]_\mu=L[X_i,\mu(X_i),\mu^2(X_i),\dots]$ over a $\mu$-difference subfield $L\subset\mathbb{K}_s$, evaluated at tuples $\underline{f}$ by replacing each $\mu^j(X_i)$ with $\mu^j(f_i)$. Because $\mu$ acts as the $p$-th power on $\mathbb{C}_\infty$, every element of $\mathbb{C}_\infty$ is transformally $\mu$-algebraic (its $\mu$-iterates are its $p$-powers), so the naive notion of difference-algebraic independence (Levin's transformal independence) is useless for arithmetic statements. Pellarin therefore isolates a class of exceptions: a nonzero $\mu$-polynomial $P$ is analytically critical if its zero variety in $\mathbb{K}_s^n$ contains a rigid analytic immersion of the unit polydisc (a convergent power-series map with everywhere invertible Jacobian), and analytically regular otherwise. Every $\mu$-polynomial of depth zero (an ordinary polynomial) is analytically regular, and so is every tame $\mu$-polynomial, one in which no exponent of any $\mu^j(X_i)$ reaches $p$. Elements $f_1,\dots,f_n\in\mathbb{K}_s$ are regularly $\mu$-independent over $L$ if $P(f_1,\dots,f_n)\ne0$ for every analytically regular $\mu$-polynomial over $L$; this implies both ordinary algebraic independence and tame $\mu$-independence, and the reverse implications are false. Finally, $L_s$ denotes the smallest $\mu$-difference subfield of $\mathbb{K}_s$ containing $K_s$ and all elements regularly $\mu$-algebraic over $K_s$ (a single element is regularly $\mu$-algebraic over $K_s$ if some nonzero analytically regular $\mu$-polynomial in one variable with coefficients in $K_s$ vanishes at it, and regularly $\mu$-transcendental otherwise); it is countable and plays the role that $\mathbb{Q}$ plays in Schanuel's conjecture. The interest of the setting is illustrated by the Anderson–Thakur function $\omega=\exp_C(\widetilde{\pi}/(\theta-t))$, the generator of the kernel of $C_{\theta-t}$ in $\mathbb{T}_1$ (where $C_{\theta-t}(b)=(\theta-t)b+\tau(b)$ is the Carlitz-type operator with $\theta$ replaced by $\theta-t$): it satisfies the tame relation $\mu^e(\omega)=(t-\theta)\omega$, so it is tamely (hence regularly) $\mu$-algebraic, while being transcendental over $K(t)$ — exactly the phenomenon the refined notion is designed to accommodate.

Problem Statement

Prove or disprove Conjecture 3.14 of Pellarin's paper "On a variant of Schanuel conjecture for the Carlitz exponential" (Conjecture 18 in the numbering of arXiv:1610.04048): for every prime power $q=p^e$, every integer $s\ge0$, every $n\ge1$, and all elements $f_1,\dots,f_n$ of the complete difference field $\mathbb{K}_s$ that are linearly independent over $\mathbb{F}_q(t_1,\dots,t_s)[\theta]$, at least $n$ among the $2n$ elements $f_1,\dots,f_n,\;\exp_C(f_1),\dots,\exp_C(f_n)$ are regularly $\mu$-independent over the countable base field $L_s$ (the smallest $\mu$-difference subfield of $\mathbb{K}_s$ containing $K_s$ and all elements regularly $\mu$-algebraic over $K_s$); that is, some $n$-element subtuple among these $2n$ elements is such that no analytically regular $\mu$-polynomial in $n$ variables with coefficients in $L_s$ vanishes at it. A complete affirmative answer includes, as the special case $s=0$, Denis' Schanuel-type conjecture for the Carlitz exponential over $\mathbb{C}_\infty$, itself open; a single explicit counterexample configuration (one $q$, one $s$, one linearly independent tuple for which every $n$-element subtuple of the $2n$ elements is regularly $\mu$-dependent over $L_s$) refutes the conjecture.

References

1. F. Pellarin, "On a variant of Schanuel conjecture for the Carlitz exponential", Journal de Théorie des Nombres de Bordeaux 29 (2017), no. 3, 845–873. DOI: 10.5802/jtnb.1004, https://doi.org/10.5802/jtnb.1004 (arXiv:1610.04048, https://arxiv.org/abs/1610.04048)
2. L. Denis, "Indépendance algébrique et exponentielle de Carlitz", Acta Arithmetica 69 (1995), no. 1, 75–89. DOI: 10.4064/aa-69-1-75-89, https://doi.org/10.4064/aa-69-1-75-89
3. L. I. Wade, "Certain quantities transcendental over GF(p^n,x)", Duke Mathematical Journal 8 (1941), 701–720. DOI: 10.1215/s0012-7094-41-00860-8, https://doi.org/10.1215/s0012-7094-41-00860-8
4. A. Thiery, "Indépendance algébrique des périodes et quasi-périodes d'un module de Drinfel'd", in: The Arithmetic of Function Fields, de Gruyter (1992), 265–284. DOI: 10.1515/9783110886153.265, https://doi.org/10.1515/9783110886153.265
5. M. Papanikolas, "Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms", Inventiones mathematicae 171 (2008), no. 1, 123–174. DOI: 10.1007/s00222-007-0073-y, https://doi.org/10.1007/s00222-007-0073-y
6. C.-Y. Chang and J. Yu, "Determination of algebraic relations among special zeta values in positive characteristic", Advances in Mathematics 216 (2007), no. 1, 321–345. DOI: 10.1016/j.aim.2007.05.012, https://doi.org/10.1016/j.aim.2007.05.012
7. G. Anderson and D. Thakur, "Tensor powers of the Carlitz module and zeta values", Annals of Mathematics (2) 132 (1990), no. 1, 159–191. DOI: 10.2307/1971503, https://doi.org/10.2307/1971503
8. B. Anglès, F. Pellarin and F. Tavares Ribeiro, "Arithmetic of positive characteristic L-series values in Tate algebras", Compositio Mathematica 152 (2016), no. 1, 1–61, appendix by F. Demeslay. DOI: 10.1112/s0010437x15007563, https://doi.org/10.1112/s0010437x15007563
9. A. Maurischat, "Prolongations of t-motives and algebraic independence of periods", Documenta Mathematica 23 (2018), 815–838. DOI: 10.25537/dm.2018v23.815-838, https://doi.org/10.25537/dm.2018v23.815-838 (arXiv:1710.08252, https://arxiv.org/abs/1710.08252)
10. O. Gezmiş, "Taelman L-values for Drinfeld modules over Tate algebras", Taiwanese Journal of Mathematics 23 (2019), no. 4. DOI: 10.1007/s40687-019-0181-5, https://doi.org/10.1007/s40687-019-0181-5
11. O. Gezmiş and F. Pellarin, "Trivial multiple zeta values in Tate algebras", International Mathematics Research Notices 2022 (2022), no. 18, 14319–14383. DOI: 10.1093/imrn/rnab104, https://doi.org/10.1093/imrn/rnab104 (arXiv:2008.07144, https://arxiv.org/abs/2008.07144)
12. K. Chung, T. Ngo Dac and F. Pellarin, "Universal families of Eulerian multiple zeta values in positive characteristic", Advances in Mathematics 422 (2023), Article 109003. DOI: 10.1016/j.aim.2023.109003, https://doi.org/10.1016/j.aim.2023.109003 (arXiv:2111.06973, https://arxiv.org/abs/2111.06973)
13. F. Pellarin, "The analytic theory of vectorial Drinfeld modular forms", Memoirs of the American Mathematical Society (2025). DOI: 10.1090/memo/1581, https://doi.org/10.1090/memo/1581 (arXiv:1910.12743, https://arxiv.org/abs/1910.12743)
14. P. Kowalski, "Positive characteristic Ax–Schanuel", Model Theory 3 (2024), 685–700. DOI: 10.2140/mt.2024.3.685, https://doi.org/10.2140/mt.2024.3.685 (arXiv:2311.00546, https://arxiv.org/abs/2311.00546)
15. D. Matsuzuki, "On algebraic independence of Taylor coefficients of certain Anderson–Thakur series", arXiv:2404.16427 (2024), https://arxiv.org/abs/2404.16427
16. T. Hasegawa, "Explicit formulas for the exponential and logarithm of the Carlitz–Tate twist, and applications", Journal de Théorie des Nombres de Bordeaux 36 (2024), no. 1, 251–291. DOI: 10.5802/jtnb.1278, https://doi.org/10.5802/jtnb.1278