# The Geometry, Algebra and Analysis of Algebraic Numbers (15w5054)

Arriving in Banff, Alberta Sunday, October 4 and departing Friday October 9, 2015

## Organizers

Francesco Amoroso (Université de Caen, Laboratoire LMNO)

Igor Pritsker (Oklahoma State University)

Christopher Smyth (University of Edinburgh)

Jeffrey Vaaler (University of Texas at Austin)

## Objectives

The meeting will present a tremendous opportunity to bring experts in many different areas together, and enable them to learn from and build on each others' specialized knowledge. Thus, on the one hand the topic of the workshop is clearly focused, while on the other hand the diversity of participants' interests will give the workshop great potential for cross-fertilisation between different areas of mathematics.

**Mahler measure and heights.**We plan to summarize recent progress and stimulate further development in this area, where the central problem is the $80$-year old Lehmer conjecture [36],[51], on the smallest Mahler measure of a nonzero noncyclotomic algebraic integer. Thus, while this fundamental problem is old and difficult, there has been steady progress towards its positive solution in recent years. Some important advances include its proofs for polynomials having a bounded number of monomials [23], with odd coefficients [16] and for algebraic numbers that generate Galois extensions of the rationals [9], generalizations of Dobrowolski-type bounds for multiplicatively independent algebraic numbers [9] and for multivariate polynomials [8], and an absolute lower bound for the height in abelian extensions [10]. The polynomials of smallest Mahler measure coming from integer symmetric matrices have also been found [24],[39]. There are many rapidly developing related directions such as computing explicit values of the Mahler measure [41], its connection with algebraic geometry and dynamics [50], [42], counting results [37], infinite fields with the Bogomolov [15] and the Northcott property [27],[20], etc. The aim of the proposed conference is to preserve and enhance this positive momentum, and to create a strong foundation for further progress of this rich area. We believe that the topic of the workshop is quite appropriate, and we hope that it will lead to solutions of important problems.

**The legacy of Schur and Siegel.**One of the aims of the meeting will be to highlight the inheritance of Schur and Siegel in this area. (Their achievements in other areas are so substantial that these have perhaps not received enough attention up to now.) One important topic is the Schur-Siegel trace problem on the smallest limit point for arithmetic means for totally real and positive algebraic integers [48], [49] ,[52],[3] . Another is the application of the Schur theory of Hardy functions to the study of Pisot numbers [19],[54], and, more generally, to the theory of nonreciprocal polynomials [5],[6]. The distribution of sets of conjugates algebraic numbers in the complex plane, whose study began with Kronecker [35], and was continued by Schur [48] and Fekete [30], is still far from being well understood. A notable exception is the area of asymptotic equidistribution for algebraic numbers of small height (or small Mahler measure), which received substantial attention in recent years (see, e.g., [13],[53],[29],[55]). The ideas of equidistribution were successfully applied in many problems, including the Schur-Siegel trace problem, cf. [46],[47].

**Relations between conjugate algebraic numbers.**The problem of how conjugate algebraic numbers may be connected algebraically is poorly understood -- the difficulty of Lehmer's problem being a consequence of this. A related problem is to find strong lower bounds for discriminants of algebraic numbers, current known exponential lower bounds all coming from consideration of field discriminants. A breakthrough on this would have consequences for Lehmer's problem.Other connections between conjugates have been studied in [11],[12] ,[25].itembf Integer Chebyshev problem and other extremal problems for polynomials with integer coefficients in various norms. The problems of minimizing norms by polynomials with integer coefficients date back to at least the work of Hilbert of 1894 [32]. They were developed by Fekete and many others [7],[17],[18],[44], but the integer Chebyshev problem remains open even in its classical setting. It has intimate connections with the distribution of conjugate algebraic integers, and with the Schur-Siegel trace problem. In the workshop, we plan to consider a range of extremal problems that have many applications in number theory.

**Applications of functional analysis.**Recent work has led to new theorems about the Weil height of algebraic numbers using techniques from functional analysis. In some cases the results can be stated in the classical language of algebraic number theory, but the proofs use a special Schauder basis for the Banach spaces identified in [4]. At present proofs using classical methods of algebraic number theory are not known. We expect that these results and techniques will be described at the workshop and become familiar to a wider audience of researchers.

**Related conferences.**There was a meeting in Paris in 1988:

*Cinquante ans des Polynômes*[1], a meeting in Banff in 2003:

*The many aspects of Mahler's measure*, and a meeting in Bristol in 2006:

*Number Theory and Polynomials} [2]*, but no others we know of. Since significant progress is intermittent and breakthroughs sparse, there is no need for frequent conferences. But now the time has come for another one. Thus the proposed meeting in 2015 is both timely and appropriate.

#### Bibliography

[1] Cinquante ans de polyônmes. Proceedings of the conference in honour of Alain Durand held in Paris, May 26--27, 1988. Edited by M. Langevin and M. Waldschmidt. Lecture Notes in Mathematics, 1415. Springer-Verlag, Berlin, 1990.

[2] Number theory and polynomials. Proceedings of the workshop held at Bristol University, Bristol, April 3--7, 2006. Edited by James McKee and Chris Smyth. London Mathematical Society Lecture Note Series, 352. Cambridge University Press, Cambridge, 2008.

[3] Aguirre, Julián; Peral, Juan Carlos. The trace problem for totally positive algebraic integers. With an appendix by Jean-Pierre Serre. London Math. Soc. Lecture Note Ser., 352, Number theory and polynomials, 1--19, Cambridge Univ. Press, Cambridge, 2008.

[4] Allcock, Daniel; Vaaler, Jeffrey D. A Banach space determined by the Weil height. Acta Arith. 136 (2009), no. 3, 279--298.

[5] Alpay, Daniel. Algorithme de Schur, espaces á noyau reproduisant et théorie des systèmes. Panoramas et Synthèses, 6. Société Mathématique de France, Paris, 1998.

[6] Alpay, Daniel. The Schur algorithm, reproducing kernel spaces and system theory. Translated from the 1998 French original by Stephen S. Wilson. SMF/AMS Texts and Monographs, 5. American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001.

[7] Amoroso, Francesco. $f$-transfinite diameter and number-theoretic applications. Ann. Inst. Fourier (Grenoble) 43 (1993), no. 4, 1179--1198.

[8] Amoroso, Francesco. On the Mahler measure in several variables. Bull. Lond. Math. Soc. 40 (2008), no. 4, 619--630.

[9] Amoroso, Francesco; David, Sinnou. Le probléme de Lehmer en dimension supérieure. J. Reine Angew. Math. 513 (1999), 145--179.

[10] Amoroso, Francesco; Zannier, Umberto. A uniform relative Dobrowolski's lower bound over abelian extensions. Bull. Lond. Math. Soc. 42 (2010), no. 3, 489--498.

[11] Baron, G.; Drmota, M.; Skal ba, M. Polynomial relations between polynomial roots, J. Algebra {177} (1995), 827--846.

[12] Berry, Neil; Dubickas, Arturas; Elkies, Noam D.; Poonen, Bjorn; Smyth, Chris . The conjugate dimension of algebraic numbers. Q. J. Math. 55 (2004), no. 3, 237--252.

[13] Y. Bilu, Limit distribution of small points on algebraic tori, Duke Math. J. 89 (1997), 465--476.

[14] Blanksby, P. E.; Montgomery, H. L. Algebraic integers near the unit circle. Acta Arith. 18 (1971), 355--369.

[15] Bombieri Enrico; Zannier Umberto. A note on heights in certain infinite extensions of $mathbb Q$. Rend. Mat. Acc. Lincei (9), 12 (2001), 5--14.

[16] Borwein, Peter; Dobrowolski, Edward; Mossinghoff, Michael J. Lehmer's problem for polynomials with odd coefficients. Ann. of Math. (2) 166 (2007), no. 2, 347--366.

[17] Borwein, Peter; Erdelyi, Tamas. The integer Chebyshev problem. Math. Comp. 65 (1996), no. 214, 661--681.

[18] Borwein, P. B.; Pritsker, I. E. The multivariate integer Chebyshev problem. Constr. Approx. 30 (2009), no. 2, 299--310.

[19] Boyd, David W. The distribution of the Pisot numbers in the real line. Séminaire de théorie des nombres, Paris 1983–84, 9–23, Progr. Math., 59, Birkhäuser Boston, Boston, MA, 1985.

[20] Checcoli, Sara; Widmer, Martin. On the Northcott property and other properties related to polynomial mappings. To appear in Math. Proc. Cambridge Philos. Soc.

[21] Deninger, Christopher. Mahler measures and Fuglede-Kadison determinants. Münster J. Math. 2 (2009), 45--63.

[22] Deninger, Christopher. Determinants on von Neumann algebras, Mahler measures and Ljapunov exponents. J. Reine Angew. Math. 651 (2011), 165--185.

[23] Dobrowolski, Edward. Mahler's measure of a polynomial in terms of the number of its monomials. Acta Arith. 123 (2006), no. 3, 201--231.

[24] Dobrowolski, Edward. A note on integer symmetric matrices and Mahler's measure. Canad. Math. Bull. 51 (2008), no. 1, 57--59.

[25] Drungilas, Paulius; Dubickas, Arturas. On subfields of a field generated by two conjugate algebraic numbers. Proc. Edinb. Math. Soc. (2) 47 (2004), no. 1, 119--123.

[26] Dubickas, Arturas. On the degree of a linear form in conjugates of an algebraic number. Illinois J. Math. 46 (2002), no. 2, 571--585.

[27] Dvornicich, Roberto; Zannier, Umberto. On the properties of Northcott and of Narkiewicz for fields of algebraic numbers. Funct. Approx. Comment. Math. 39 (2008), part 1, 163--173.

[28] Everest, Graham; Ward, Thomas. Heights of polynomials and entropy in algebraic dynamics. Universitext. Springer-Verlag London, Ltd., London, 1999.

[29] C. Favre and J. Rivera-Letelier, Equidistribution quantitative des points de petite hauteur sur la droite projective, Math. Ann. 335 (2006), 311--361; Corrigendum in Math. Ann. 339 (2007), 799--801.

[30] Fekete, M. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17 (1923), no. 1, 228--249.

[31] Flammang, V. On the absolute trace of polynomials having all zeros in a sector. Experiment. Math. 17 (2008), no. 4, 443--450.

[32] Hilbert, D. Ein Beitrag zur Theorie des Legendreschen Polynoms, Acta Math. 18 (1894), 155-159.

[33] Hironaka, Eriko, On hyperbolic perturbations of algebraic links and small Mahler measure. Singularities in geometry and topology 2004, 77–94, Adv. Stud. Pure Math., 46, Math. Soc. Japan, Tokyo, 2007.

[34] Kellerhals, Ruth, Cofinite hyperbolic Coxeter groups, minimal growth rate and Pisot numbers. Algebr. Geom. Topol. 13 (2013), no. 2, 1001–1025.

[35] Kronecker, L. Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173--175.

[36] Lehmer, D. H. Factorization of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), no. 3, 461--479.

[37] Masser, David; Vaaler, Jeffrey D. Counting algebraic numbers with large height. I. Diophantine approximation, 237--243, Dev. Math., 16, Springer, Wien, New York, Vienna, 2008.

[38] McKee, James; Smyth, Chris. Salem numbers, Pisot numbers, Mahler measure, and graphs. Experiment. Math. 14 (2005), no. 2, 211--229.

[39] McKee, James; Smyth, Chris. Integer symmetric matrices of small spectral radius and small Mahler measure. Int. Math. Res. Not. IMRN 2012, no. 1, 102--136.

[40] Montgomery, Hugh L. Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, 84. American Mathematical Society, Providence, RI, 1994.

[41] Mossinghoff, Michael J.; Rhin, Georges; Wu, Qiang. Minimal Mahler measures. Experiment. Math. 17 (2008), no. 4, 451--458.

[42] Ostafe, Alina; Shparlinski, Igor E. On the degree growth in some polynomial dynamical systems and nonlinear pseudorandom number generators. Math. Comp. 79 (2010), no. 269, 501--511.

[43] Pritsker, Igor E. The Gelfond-Schnirelman method in prime number theory. Canad. J. Math. 57 (2005), no. 5, 1080--1101.

[44] Pritsker, Igor E. Small polynomials with integer coefficients. J. Anal. Math. 96 (2005), 151--190.

[45] Pritsker, Igor E. Polynomial inequalities, Mahler's measure, and multipliers. Number theory and polynomials, 255--276, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008.

[46] I. E. Pritsker, Means of algebraic numbers in the unit disk. C. R. Math. Acad. Sci. Paris 347 (2009), no. 3-4, 119--122.

[47] I. E. Pritsker, Distribution of algebraic numbers, J. Reine Angew. Math. 657 (2011), 57--80.

[48] Schur, I. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 1, 377-402 (1918).

[49] Siegel, Carl Ludwig. The trace of totally positive and real algebraic integers. Ann. of Math. (2) 46 (1945), 302--312.

[50] Silverman, Joseph H. The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241. Springer, New York, 2007.

[51] Smyth, Chris. The Mahler measure of algebraic numbers: a survey. Number theory and polynomials, 322--349, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge, 2008.

[52] C. J. Smyth, The mean values of totally real algebraic integers, Math. Comp. 42 (1984), 663--681.

[53] L. Szpiro, E. Ullmo, S. Zhang, Equir'epartition des petits points, Invent. Math. 127 (1997), 337--347.

[54] Thurnheer, Peter. Kleine Pisot-Vijayaraghavan-Zahlen und die Fibonacci-Folge. Monatsh. Math. 95 (1983), no. 4, 321--331.

[55] Zhang, Shou-Wu, Equidistribution of small points on abelian varieties. Ann. of Math. (2) 147 (1998), no. 1, 159–165.