Publications

Journal Articles

2025. The field of iterates of a rational function. J. Théor. Nombres Bordeaux, 37(2): 691–709. Francesco Veneziano and Solomon Vishkautsan.
doi arXiv
We study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with the property that the field of definition of one of their iterates drops in degree (over a given base field). We show with families of examples that this characterization does not hold for rational functions. Finally, we also classify fractional linear transformations with this property.
```
@article{MR4963751,
  author = {Veneziano, Francesco and Vishkautsan, Solomon},
  title = {The field of iterates of a rational function},
  journal = {J. Th\'{e}or. Nombres Bordeaux},
  fjournal = {Journal de Th\'{e}orie des Nombres de Bordeaux},
  volume = {37},
  year = {2025},
  number = {2},
  pages = {691--709},
  issn = {1246-7405,2118-8572},
  mrclass = {37P05 (11C08 11R04 37P15)},
  mrnumber = {4963751},
  doi = {10.5802/jtnb.1337},
  url = {https://doi.org/10.5802/jtnb.1337},
  eprint = {2404.04939},
  eprinttype = {arxiv},
  eprintclass = {math.NT}
}
```

2025. Classification of rational angles in plane lattices II. Math. Ann., 393(3-4): 3713–3771. Roberto Dvornicich, Davide Lombardo, Francesco Veneziano, and Umberto Zannier.

doi

This paper is a continuation of an earlier one, and completes a classification of the configurations of points in a plane lattice that determine angles that are rational multiples of $\pi$. We give a complete and explicit description of lattices according to which of these configurations can be found among their points.

@article{MR5007552,
  author = {Dvornicich, Roberto and Lombardo, Davide and Veneziano, Francesco and Zannier, Umberto},
  title = {Classification of rational angles in plane lattices {II}},
  journal = {Math. Ann.},
  fjournal = {Mathematische Annalen},
  volume = {393},
  year = {2025},
  number = {3-4},
  pages = {3713--3771},
  issn = {0025-5831,1432-1807},
  mrclass = {11D61 (11H06 14G05 51M05)},
  mrnumber = {5007552},
  doi = {10.1007/s00208-025-03214-6},
  url = {https://doi.org/10.1007/s00208-025-03214-6}
}

2022. Classification of rational angles in plane lattices. Bull. Amer. Math. Soc. (N.S.), 59(2): 191–226. Roberto Dvornicich, Francesco Veneziano, and Umberto Zannier.
doi arXiv
This paper is concerned with configurations of points in a plane lattice which determine angles that are rational multiples of $\pi$. We shall study how many such angles may appear in a given lattice and in which positions, allowing the lattice to vary arbitrarily. This classification turns out to be much less simple than could be expected, leading even to parametrizations involving rational points on certain algebraic curves of positive genus.
```
@article{MR4390499,
  author = {Dvornicich, Roberto and Veneziano, Francesco and Zannier, Umberto},
  title = {Classification of rational angles in plane lattices},
  year = {2022},
  volume = {59},
  number = {2},
  pages = {191--226},
  issn = {0273-0979},
  mrclass = {11D61 (11G30 11H06 14G05 51M05)},
  mrnumber = {4390499},
  journal = {Bull. Amer. Math. Soc. (N.S.)},
  fjournal = {American Mathematical Society. Bulletin. New Series},
  eprint = {2005.13598},
  eprinttype = {arxiv},
  eprintclass = {math.NT},
  doi = {10.1090/bull/1723},
  url = {https://doi.org/10.1090/bull/1723}
}
```
2022. Finiteness and periodicity of continued fractions over quadratic number fields. Bull. Soc. Math. France, 150(1): 77–109. Zuzana Masáková, Tomáš Vávra, and Francesco Veneziano.
doi arXiv
In this paper we prove a periodicity theorem for certain continued fractions with partial quotients in the ring of integers of a fixed quadratic field. This theorem generalizes the classical theorem of Lagrange to a large set of continued fraction expansions. As an application we consider the $\beta$-continued fractions and show that for any quadratic Perron number $\beta$, the $\beta$-continued fraction expansion of elements in $\mathbb{Q}(\beta)$ is either finite of eventually periodic. More in general we examine the finiteness and periodicity of the $\beta$-continued fractions for all quadratic integers $\beta$, thus studying problems raised by Rosen and Bernat.
```
@article{MR4440296,
  author = {Mas\'{a}kov\'{a}, Zuzana and V\'{a}vra, Tom\'{a}\v{s} and Veneziano, Francesco},
  title = {Finiteness and periodicity of continued fractions over
                quadratic number fields},
  journal = {Bull. Soc. Math. France},
  fjournal = {Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France},
  volume = {150},
  year = {2022},
  number = {1},
  pages = {77--109},
  issn = {0037-9484},
  mrclass = {11A55 (11J70 11K16)},
  mrnumber = {4440296},
  doi = {10.24033/bsmf.2845},
  url = {https://doi.org/10.24033/bsmf.2845},
  eprint = {1911.07670},
  eprinttype = {arxiv},
  eprintclass = {math.NT}
}
```

2022. On the integral values of a curious recurrence. Riv. Math. Univ. Parma (N.S.), 13(1): 1–18. Roberto Dvornicich, Francesco Veneziano, and Umberto Zannier.

url

We discuss an elementary problem, initially proposed for the Romanian Mathematical Olympiad, which leads to interesting remarks of various nature. We relate the problem to the theory of linear recurrence sequences with non-constant coefficients and their $p$-adic behaviour. Our considerations can be applied to a larger set of similarly-defined recurrences.

@article{MR4456578,
  author = {Dvornicich, Roberto and Veneziano, Francesco and Zannier, Umberto},
  title = {On the integral values of a curious recurrence},
  journal = {Riv. Math. Univ. Parma (N.S.)},
  fjournal = {Rivista di Matematica della Universit\`a di Parma. New Series. A
                Journal of Pure and Applied Mathematics},
  volume = {13},
  year = {2022},
  number = {1},
  pages = {1--18},
  issn = {0035-6298},
  mrclass = {11B37 (11B83)},
  mrnumber = {4456578},
  url = {https://www.rivmat.unipr.it/vols/2022-13-1/01-dvornicich.html}
}

2022. Hyperelliptic continued fractions in the singular case of genus zero. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 33(4): 795–832. Francesco Ballini and Francesco Veneziano.
doi arXiv
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier that the sequence of the degrees of the partial quotients of the continued fraction expansion of $\sqrt{D(t)}$ is always periodic, even when the expansion itself is not. In this article we work out in detail the case in which the curve $y^2=D(t)$ has genus 0, establishing explicit geometric conditions corresponding to the appearance of partial quotients of certain degrees in the continued fraction expansion. We also show that there are non-trivial polynomials $D(t)$ with non-periodic expansions such that infinitely many partial quotients have degree greater than one.
```
@article{MR4595292,
  author = {Ballini, Francesco and Veneziano, Francesco},
  title = {Hyperelliptic continued fractions in the singular case of
                genus zero},
  journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
  fjournal = {Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni},
  volume = {33},
  year = {2022},
  number = {4},
  pages = {795--832},
  issn = {1120-6330,1720-0768},
  mrclass = {11J70 (11A55 14H50 40A15)},
  mrnumber = {4595292},
  doi = {10.4171/rlm/989},
  url = {https://doi.org/10.4171/rlm/989},
  eprint = {2108.06560},
  eprinttype = {arxiv},
  eprintclass = {math.NT}
}
```
2021. Explicit height bounds for $K$-rational points on transverse curves in powers of elliptic curves. Pacific J. Math., 315(2): 477–503. Francesco Veneziano and Evelina Viada.
doi arXiv
Let $C$ be an algebraic curve embedded transversally in a power $E^N$ of an elliptic curve $E$ with complex multiplication. We produce a good explicit bound for the height of all the algebraic points on $C$ contained in the union of all proper algebraic subgroups of $E^N$. The method gives a totally explicit version of the Manin–Demjanenko theorem in the elliptic case and complements previous results only proved when $E$ does not have complex multiplication.
```
@article{MR4366750,
  author = {Veneziano, Francesco and Viada, Evelina},
  title = {Explicit height bounds for {$K$}-rational points on transverse
                curves in powers of elliptic curves},
  journal = {Pacific J. Math.},
  fjournal = {Pacific Journal of Mathematics},
  volume = {315},
  year = {2021},
  number = {2},
  pages = {477--503},
  issn = {0030-8730},
  mrclass = {11G50 (14G40)},
  mrnumber = {4366750},
  mrreviewer = {Gabriel Andreas Dill},
  doi = {10.2140/pjm.2021.315.477},
  url = {https://doi.org/10.2140/pjm.2021.315.477},
  eprint = {2006.02538},
  eprinttype = {arxiv},
  eprintclass = {math.NT}
}
```

2020. A note on the zeroes of the Fredholm series. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 31(4): 651–666. Umberto Zannier.

doi arXiv

The issue had been raised whether the function $z+z^2+\dotsb+z^{2^n}+\dotsb$, sometimes called Fredholm series, has infinitely many zeroes in the unit disk. We provide an affirmative answer, proving that in fact every complex number occurs as a value infinitely many times, even restricting the function to any open set meeting the unit circle.

@article{MR4215674,
  author = {Zannier, Umberto},
  title = {A note on the zeroes of the {Fredholm} series},
  titleaddon = {With an appendix by Francesco Veneziano},
  journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
  fjournal = {Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni},
  volume = {31},
  year = {2020},
  number = {4},
  pages = {651--666},
  issn = {1120-6330},
  mrclass = {30B10 (30B30)},
  mrnumber = {4215674},
  doi = {10.4171/rlm/909},
  url = {https://doi.org/10.4171/rlm/909},
  eprint = {2006.11922},
  eprinttype = {arxiv},
  eprintclass = {math.CV}
}

2019. An effective criterion for periodicity of $\ell$-adic continued fractions. Math. Comp., 88(318): 1851–1882. Laura Capuano, Francesco Veneziano, and Umberto Zannier.
doi
The theory of continued fractions has been generalized to $\ell$-adic numbers by several authors and presents many differences with respect to the real case. In the present paper we investigate the expansion of rationals and quadratic irrationals for the l-adic continued fractions introduced by Ruban. In this case, rational numbers may have a periodic non-terminating continued fraction expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange’s theorem holds. We give general explicit criteria to establish the periodicity of the expansion in both the rational and the quadratic case (for rationals, the qualitative result is due to Laohakosol.
```
@article{MR3925488,
  author = {Capuano, Laura and Veneziano, Francesco and Zannier, Umberto},
  fjournal = {Mathematics of Computation},
  issn = {0025-5718},
  journal = {Math. Comp.},
  mrclass = {11J70 (11D88 11Y16)},
  mrnumber = {3925488},
  mrreviewer = {Jeffrey O. Shallit},
  number = {318},
  pages = {1851--1882},
  title = {An effective criterion for periodicity of {$\ell$}-adic continued fractions},
  doi = {10.1090/mcom/3385},
  url = {https://doi.org/10.1090/mcom/3385},
  volume = {88},
  year = {2019}
}
```
2019. The explicit Mordell conjecture for families of curves. Forum Math. Sigma, 7: Paper No. e31, 62. Sara Checcoli, Francesco Veneziano, and Evelina Viada. With an appendix by M. Stoll.
doi
In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin-Demjanenko and the analysis of our explicit examples is carried to conclusion.
```
@article{MR4010563,
  author = {Checcoli, Sara and Veneziano, Francesco and Viada, Evelina},
  fjournal = {Forum of Mathematics. Sigma},
  journal = {Forum Math. Sigma},
  mrclass = {11G50 (11G30 11Y50 14G40)},
  mrnumber = {4010563},
  mrreviewer = {Robin de Jong},
  pages = {Paper No. e31, 62},
  eid = {e31},
  pagetotal = {62},
  title = {The explicit {Mordell} conjecture for families of curves},
  note = {With an appendix by M. Stoll},
  doi = {10.1017/fms.2019.20},
  url = {https://doi.org/10.1017/fms.2019.20},
  volume = {7},
  year = {2019}
}
```
2018. Pisot unit generators in number fields. J. Symbolic Comput., 89: 94–108. Tomáš Vávra and Francesco Veneziano.
doi
Pisot numbers are real algebraic integers bigger than 1, whose other conjugates all have modulus smaller than 1. In this paper we deal with the algorithmic problem of finding the smallest Pisot unit generating a given number field. We first solve this problem in all real fields, then we consider the analogous problem involving the so called complex Pisot numbers and we solve it in all number fields that admit such a generator, in particular this includes all fields without CM.
```
@article{MR3804808,
  author = {{V\'avra}, Tom{\'a}{\v s} and Veneziano, Francesco},
  fjournal = {Journal of Symbolic Computation},
  issn = {0747-7171},
  journal = {J. Symbolic Comput.},
  mrclass = {11K16},
  mrnumber = {3804808},
  mrreviewer = {Gerem\'{\i}as Polanco},
  pages = {94--108},
  title = {{Pisot} unit generators in number fields},
  doi = {10.1016/j.jsc.2017.11.005},
  url = {https://doi.org/10.1016/j.jsc.2017.11.005},
  volume = {89},
  year = {2018}
}
```
2017. On the explicit Torsion Anomalous Conjecture. Trans. Amer. Math. Soc., 369(9): 6465–6491. Sara Checcoli, Francesco Veneziano, and Evelina Viada.
doi
The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a totally explicit bound for the Néron-Tate height of all maximal $V$-torsion anomalous points of relative codimension one in the non-CM case, and an analogous effective result in the CM case. As an application, we obtain the finiteness of such points. In addition, we deduce some new explicit results in the context of the effective Mordell-Lang Conjecture; in particular we bound the Néron-Tate height of the rational points of an explicit family of curves of increasing genus.
```
@article{MR3660229,
  author = {Checcoli, Sara and Veneziano, Francesco and Viada, Evelina},
  fjournal = {Transactions of the American Mathematical Society},
  issn = {0002-9947},
  journal = {Trans. Amer. Math. Soc.},
  mrclass = {11G50 (11G05 14G40)},
  mrnumber = {3660229},
  mrreviewer = {Yu Yasufuku},
  number = {9},
  pages = {6465--6491},
  title = {On the explicit Torsion Anomalous Conjecture},
  doi = {10.1090/tran/6893},
  url = {https://doi.org/10.1090/tran/6893},
  volume = {369},
  year = {2017}
}
```
2014. On torsion anomalous intersections. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 25(1): 1–36. Sara Checcoli, Francesco Veneziano, and Evelina Viada.
doi
A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^{ta}$ of all $V$-torsion anomalous varieties is open and dense in $V$. We prove some cases of this conjecture. We show that the $V$-torsion anomalous varieties of relative codimension one are non-dense in any weak-transverse variety $V$ embedded in a product of elliptic curves with CM. We give explicit uniform bounds in the dependence on $V$. As an immediate consequence we prove the conjecture for $V$ of codimension two in a product of CM elliptic curves. We also point out some implications on the effective Mordell-Lang Conjecture.
```
@article{MR3180478,
  author = {Checcoli, Sara and Veneziano, Francesco and Viada, Evelina},
  fjournal = {Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni},
  issn = {1120-6330},
  journal = {Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.},
  mrclass = {11G50 (14G40)},
  mrnumber = {3180478},
  mrreviewer = {Dimitrios Poulakis},
  number = {1},
  pages = {1--36},
  title = {On torsion anomalous intersections},
  doi = {10.4171/RLM/666},
  url = {https://doi.org/10.4171/RLM/666},
  volume = {25},
  year = {2014}
}
```
2012. A sharp Bogomolov-type bound. New York J. Math., 18: 891–910. Sara Checcoli, Francesco Veneziano, and Evelina Viada.
url
We prove a sharp lower bound for the essential minimum of a nontranslate variety in certain abelian varieties. This uses and generalises a result of Galateau. Our bound is a new step in the direction of an abelian analogue by David and Philippon of a toric conjecture of Amoroso and David and has applications in the framework of anomalous intersections.
```
@article{MR2991428,
  author = {Checcoli, Sara and Veneziano, Francesco and Viada, Evelina},
  fjournal = {New York Journal of Mathematics},
  journal = {New York J. Math.},
  mrclass = {11J95 (11G10 11G50)},
  mrnumber = {2991428},
  mrreviewer = {Aur{\'e}lien Galateau},
  pages = {891--910},
  title = {A sharp {Bogomolov}-type bound},
  url = {http://nyjm.albany.edu:8000/j/2012/18_891.html},
  volume = {18},
  year = {2012}
}
```
2011. Quadratic Integral Solutions to Double Pell. Rend. Semin. Mat. Univ. Padova, 126: 47–61. Francesco Veneziano.
doi
We study the quadratic integral points—that is, ($S$-)integral points defined over any extension of degree two of the base field—on a curve defined in $\mathbb{P}^3$ by a system of two Pell equations. Such points belong to three families explicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field, the equations defining the curve and the set $S$. We exploit the peculiar geometry of the curve to adapt the proof of a theorem of Vojta, which in this case does not apply.
```
@article{MR2918198,
  author = {Veneziano, Francesco},
  fjournal = {Rendiconti del Seminario Matematico della Universit{\`a} di Padova. Mathematical Journal of the University of Padua},
  isbn = {978-88-7784-335-7},
  issn = {0041-8994},
  journal = {Rend. Semin. Mat. Univ. Padova},
  mrclass = {11D09},
  mrnumber = {2918198},
  mrreviewer = {Hizuru Yamagishi},
  pages = {47--61},
  title = {Quadratic Integral Solutions to Double {Pell} Equations},
  doi = {10.4171/RSMUP/126-3},
  url = {https://doi.org/10.4171/RSMUP/126-3},
  volume = {126},
  year = {2011}
}
```

Book Chapters

2019. Hyperelliptic Continued Fractions and Generalized Jacobians. In Clemens Fuchs and Wüstholz Gisbert, editors, Arithmetic and Geometry: Ten Years in Alpbach, Annals of Mathematics Studies, pages 56–101. Princeton University Press, Princeton, NJ, edition. Laura Capuano, Peter Jossen, Christina Karolus, and Francesco Veneziano.

doi

@incollection{AlpbachChapterCJKV,
  author = {Capuano, Laura and Jossen, Peter and Karolus, Christina and Veneziano, Francesco},
  title = {Hyperelliptic Continued Fractions and Generalized {Jacobians}},
  subtitle = {Minicourse Given by {Umberto} {Zannier}},
  booktitle = {Arithmetic and Geometry: Ten Years in Alpbach},
  booktitleaddon = {AMS-202},
  series = {Annals of Mathematics Studies},
  number = {202},
  editor = {Fuchs, Clemens and Gisbert, W{\"u}stholz},
  type = {chapter},
  chapter = {3},
  pages = {56--101},
  isbn = {9780691193786},
  publisher = {Princeton University Press, Princeton, NJ},
  year = {2019},
  mrclass = {11G30 (11A55 14)},
  mrnumber = {4353100},
  doi = {https://doi.org/10.2307/j.ctvfrxrcd.6},
  url = {http://www.jstor.org/stable/j.ctvfrxrcd.6}
}

Conference proceedings

August 2024. Learning to Play 7 Wonders Duel Without Human Supervision. In 2024 IEEE Conference on Games (CoG), pages 1–4. August. Giovanni Paolini, Lorenzo Moreschini, Francesco Veneziano, and Alessandro Iraci.

doi arXiv

This paper introduces ZeusAI, an artificial intelligence system developed to play the board game 7 Wonders Duel. Inspired by the AlphaZero reinforcement learning algorithm, ZeusAI relies on a combination of Monte Carlo Tree Search and a Transformer Neural Network to learn the game without human supervision. ZeusAI competes at the level of top human players, develops both known and novel strategies, and allows us to test rule variants to improve the game’s balance. This work demonstrates how AI can help in understanding and enhancing board games.

@inproceedings{ProceedingsZeusAI,
  author = {Paolini, Giovanni and Moreschini, Lorenzo and Veneziano, Francesco and Iraci, Alessandro},
  booktitle = {2024 IEEE Conference on Games (CoG)},
  title = {Learning to Play 7 Wonders Duel Without Human Supervision},
  year = {2024},
  month = aug,
  volume = {},
  number = {},
  pages = {1--4},
  eventtitle = {2024 IEEE Conference on Games},
  eventdate = {2024-08-05},
  venue = {Milano, Italy},
  keywords = {Monte Carlo methods;Neural networks;Games;Reinforcement learning;Learning (artificial intelligence);Transformers;Artificial Intelligence;Reinforcement Learning;7 Wonders Duel;Board Game Strategy;Board Game Enhancement},
  issn = {2325-4289},
  doi = {10.1109/CoG60054.2024.10645555},
  eprint = {2406.00741},
  archiveprefix = {arXiv},
  eprinttype = {arxiv},
  primaryclass = {cs.AI},
  eprintclass = {cs.AI}
}

2018. An effective criterion for periodicity of $\ell$-adic continued fractions. In Numeration 2018, pages 73–74. Francesco Veneziano.

url

@inproceedings{ProceedingsNumeration,
  author = {Veneziano, Francesco},
  title = {An effective criterion for periodicity of $\ell$-adic continued fractions},
  booktitle = {Numeration 2018},
  pages = {73--74},
  year = {2018},
  eventtitle = {Numeration 2018},
  eventdate = {2018-05-22},
  venue = {Universit{\'e} Paris Diderot},
  url = {https://numeration2018.sciencesconf.org/data/pages/num18_abstracts.pdf}
}

2011. Punti interi quadratici su curve di genere 1 definite da una doppia equazione di Pell. In Atti del XIX Congresso dell'Unione Matematica Italiana, page 946. Unione Matematica Italiana. Francesco Veneziano.

url

@inproceedings{ProceedingsUMI,
  author = {Veneziano, Francesco},
  title = {Punti interi quadratici su curve di genere 1 definite da una doppia equazione di {Pell}},
  booktitle = {Atti del XIX Congresso dell'Unione Matematica Italiana},
  pages = {946},
  year = {2011},
  eventtitle = {XIX Congresso dell'Unione Matematica Italiana},
  eventdate = {2011-09-12},
  organization = {Unione Matematica Italiana},
  venue = {University of Bologna},
  language = {Italian},
  url = {http://umi.dm.unibo.it/umi/congressi/}
}

Preprints

2026. Achieving Expert-Level Performance in 7 Wonders Duel Through Deep Reinforcement Learning. Giovanni Paolini, Lorenzo Moreschini, Francesco Veneziano, and Alessandro Iraci.
2025. On the disk of convergence of algebraic power series. Francesco Veneziano and Umberto Zannier.
arXiv
2025. $p$-adically convergent loci in varieties arising from periodic continued fractions. Laura Capuano, Marzio Mula, Lea Terracini, and Francesco Veneziano.
arXiv