Geodesic
Newman cyclotomic polynomials, refinable splines and the Euler binary partition function
Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1783-1802 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The class of cyclotomic polynomials (integer polynomials that have primitive complex roots of unity as their roots) is well studied in the literature. We show that its subclass, $k$-cyclotomic polynomials ($k\geqslant 2$) for which the orders of all complex roots have a common divisor $k$, possesses some remarkable properties. Such polynomials generate refinable splines, describe the asymptotic growth of the Euler binary partition function, and so on. Moreover, $k$-cyclotomic polynomials can efficiently be recognized by means of their $k$-Toeplitz matrices. Special attention is paid to $k$-cyclotomic Newman (0-1) polynomials, for which we identify one particular family. We prove that all $k$-cyclotomic polynomials are divisors of polynomials in this family and conjecture that they all actually belong to that family. As an application, we sharpen the asymptotics of the Euler binary partition function and find an explicit formula for it in the case of regular growth. Bibliography: 49 titles.
Keywords: Newman polynomial, spline
Mots-clés : refinement equations, cyclotomic polynomial. 
@article{SM_2018_209_12_a5,
     author = {V. Yu. Protasov and Ya. Wang},
     title = {Newman cyclotomic polynomials, refinable splines and the {Euler} binary partition function},
     journal = {Sbornik. Mathematics},
     pages = {1783--1802},
     year = {2018},
     volume = {209},
     number = {12},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_12_a5/}
}
TY  - JOUR
AU  - V. Yu. Protasov
AU  - Ya. Wang
TI  - Newman cyclotomic polynomials, refinable splines and the Euler binary partition function
JO  - Sbornik. Mathematics
PY  - 2018
SP  - 1783
EP  - 1802
VL  - 209
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/SM_2018_209_12_a5/
LA  - en
ID  - SM_2018_209_12_a5
ER  - 
%0 Journal Article
%A V. Yu. Protasov
%A Ya. Wang
%T Newman cyclotomic polynomials, refinable splines and the Euler binary partition function
%J Sbornik. Mathematics
%D 2018
%P 1783-1802
%V 209
%N 12
%U http://geodesic.mathdoc.fr/item/SM_2018_209_12_a5/
%G en
%F SM_2018_209_12_a5
V. Yu. Protasov; Ya. Wang. Newman cyclotomic polynomials, refinable splines and the Euler binary partition function. Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1783-1802. http://geodesic.mathdoc.fr/item/SM_2018_209_12_a5/

[1] A. Abdollahi, J. Cheshmavar, M. Taghavi, “Wavelets generated by the Rudin–Shapiro polynomials”, Cent. Eur. J. Math., 9:2 (2011), 441–448 | DOI | MR | Zbl

[2] J. Ablinger, J. Blümlein, C. Schneider, “Harmonic sums and polylogarithms generated by cyclotomic polynomials”, J. Math. Phys., 52:10 (2011), 102301, 52 pp. | DOI | MR | Zbl

[3] K. Anders, M. Dennison, J. W. Lansing, B. Reznick, “Congruence properties of binary partition functions”, Ann. Comb., 17:1 (2013), 15–26 | DOI | MR | Zbl

[4] T. M. Apostol, “Resultants of cyclotomic polynomials”, Proc. Amer. Math. Soc., 24:3 (1970), 457–462 | DOI | MR | Zbl

[5] E. Bach, J. Shallit, “Factoring with cyclotomic polynomials”, Math. Comput., 52:185 (1989), 201–219 | DOI | MR | Zbl

[6] L. Berg, G. Plonka, “Some notes on two-scale difference equations”, Functional equations and inequalities, Math. Appl., 518, Kluwer Acad. Publ., Dordrecht, 2000, 7–29 | MR | Zbl

[7] P. Borwein, K.-K. S. Choi, “On cyclotomic polynomials with $\pm 1$ coefficients”, Experiment. Math., 8:4 (1999), 399–407 | DOI | MR | Zbl

[8] A. Bettayeb, T. Goodman, “Some properties of multi-box splines”, J. Approx. Theory, 163:2 (2011), 197–212 | DOI | MR | Zbl

[9] N. G. de Bruijn, “On Mahler's partition problem”, Nederl. Akad. Wetensch., Proc., 51 (1948), 659–669, = Indagationes Math. 10, 210–220 (1948) | MR | Zbl

[10] L. Carlitz, “Generating functions and partition problems”, Proc. Sympos. Pure Math., 8, Amer. Math. Soc., Providence, R.I., 1965, 144–169 | DOI | MR | Zbl

[11] A. S. Cavaretta, W. Dahmen, C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93, no. 453, Amer. Math. Soc., Providence, R.I., 1991, vi+186 pp. | DOI | MR | Zbl

[12] R. F. Churchhouse, “Congruence properties of the binary partition function”, Proc. Cambridge Philos. Soc., 66:2 (1969), 371–376 | DOI | MR | Zbl

[13] E. A. Ciolan, P. A. García-Sánchez, P. Moree, “Cyclotomic numerical semigroups”, SIAM J. Discrete Math., 30:2 (2016), 650–668 | DOI | MR | Zbl

[14] Xin-Rong Dai, De-Jun Feng, Yang Wang, “Classification of refinable splines”, Constr. Approx., 24:2 (2006), 187–200 | DOI | MR | Zbl

[15] Xin-Rong Dai, De-Jun Feng, Yang Wang, “Structure of refinable splines”, Appl. Comput. Harmon. Anal., 22:3 (2007), 374–381 | DOI | MR | Zbl

[16] Xin-Rong Dai, Yang Wang, “Classification of refinable splines in $\mathbb R^d$”, Constr. Approx., 31:3 (2010), 343–358 | DOI | MR | Zbl

[17] H. Davenport, Multiplicative number theory, Grad. Texts in Math., 74, 2nd ed., rev. by H. L. Montgomery, Springer-Verlag, New York–Berlin, 1980, xiii+177 pp. | DOI | MR | Zbl

[18] G. Dresden, “Resultants of cyclotomic polynomials”, Rocky Mountain J. Math., 42:5 (2012), 1461–1469 | DOI | MR | Zbl

[19] A. Dubickas, “The divisors of Newman polynomials”, Fiz. Mat. Fak. Moksl. Semin. Darb., 6 (2003), 25–28 | MR | Zbl

[20] J. M. Dumont, N. Sidorov, A. Thomas, “Number of representations related to a linear recurrent basis”, Acta Arith., 88:4 (1999), 371–396 | DOI | MR | Zbl

[21] L. Euler, Introductio in analysin infinitorum (Lausanne, 1798), Opera Omnia. Series Prima: Opera Math., 8, B. G. Teubner, Leipzig, 1922, xii+392 pp. | MR | MR | Zbl | Zbl

[22] De-Jun Feng, P. Liardet, A. Thomas, “Partition functions in numeration systems with bounded multiplicity”, Unif. Distrib. Theory, 9:1 (2014), 43–77 | MR | Zbl

[23] De-Jun Feng, N. Sidorov, “Growth rate for beta-expansions”, Monatsh. Math., 162:1 (2011), 41–60 | DOI | MR | Zbl

[24] C.-E. Fröberg, “Accurate estimation of the number of binary partitions”, Nordisk Tidskr. Informationsbehandling (BIT), 17:4 (1977), 386–391 | DOI | MR | Zbl

[25] K. G. Hare, “Base-$d$ expansions with digits $0$ to $q-1$”, Exp. Math., 24:3 (2015), 295–303 | DOI | MR | Zbl

[26] M. J. Hirn, “The refinability of step functions”, Proc. Amer. Math. Soc., 136:3 (2008), 899–908 | DOI | MR | Zbl

[27] N. Guglielmi, V. Protasov, “Exact computation of joint spectral characteristics of linear operators”, Found. Comput. Math., 13:1 (2013), 37–97 | DOI | MR | Zbl

[28] T. N. T. Goodman, “Characterising pairs of refinable splines”, J. Approx. Theory, 91:3 (1997), 368–385 | DOI | MR | Zbl

[29] D. E. Knuth, “An almost linear recurrence”, Fibonacci Quart., 4 (1966), 117–128 | MR | Zbl

[30] R. P. Kurshan, A. M. Odlyzko, “Values of cyclotomic polynomials at roots of unity”, Math. Scand., 49:1 (1981), 15–35 | DOI | MR | Zbl

[31] S. Lang, Cyclotomic fields I and II, With an appendix by K. Rubin, Grad. Texts in Math., 121, Combined 2nd ed., Springer-Verlag, New York, 1990, xviii+433 pp. | DOI | MR | Zbl

[32] W. Lawton, S. L. Lee, Zuowei Shen, “Characterization of compactly supported refinable splines”, Adv. Comput. Math., 3:1-2 (1995), 137–145 | DOI | MR | Zbl

[33] M. Latapy, “Partitions of an integer into powers”, Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, 215–227 | MR | Zbl

[34] K. Mahler, “On a special functional equation”, J. London Math. Soc., 15 (1940), 115–123 | DOI | MR | Zbl

[35] C. A. Micchelli, A. Pinkus, “Descartes systems from corner cutting”, Constr. Approx., 7:2 (1991), 161–194 | DOI | MR | Zbl

[36] G. Molteni, “Representation of a $2$-power as sum of $k$ $2$-powers: the asymptotic behavior”, Int. J. Number Theory, 8:8 (2012), 1923–1963 | DOI | MR | Zbl

[37] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Wavelet theory, Transl. Math. Monogr., 239, Amer. Math. Soc., Providence, RI, 2011, xiv+506 pp. | MR | MR | Zbl | Zbl

[38] W. B. Pennington, “On Mahler's partition problem”, Ann. of Math. (2), 57:3 (1953), 531–546 | DOI | MR | Zbl

[39] J. L. Pfaltz, “Evaluating the binary partition function when $N=2^n$”, Congr. Numer., 109 (1995), 3–12 | MR | Zbl

[40] V. Yu. Protasov, “Asymptotic behaviour of the partition function”, Sb. Math., 191:3 (2000), 381–414 | DOI | DOI | MR | Zbl

[41] V. Yu. Protasov, “On the asymptotics of the binary partition function”, Math. Notes, 76:1 (2004), 144–149 | DOI | DOI | MR | Zbl

[42] V. Yu. Protasov, “Piecewise-smooth refinable functions”, St. Petersburg Math. J., 16:5 (2005), 821–835 | DOI | MR | Zbl

[43] V. Yu. Protasov, “Spectral factorization of 2-block Toeplitz matrices and refinement equations”, St. Petersburg Math. J., 18:4 (2007), 607–646 | DOI | MR | Zbl

[44] V. Yu. Protasov, “The Euler binary partition function and subdivision schemes”, Math. Comp., 86:305 (2017), 1499–1524 | DOI | MR | Zbl

[45] B. Reznick, “Some binary partition functions”, Analytic number theory, Proceedings of a conference in honor of P. T. Bateman, B. C. Berndt, H. G. Diamond, H. Halberstam, A. Hildebrand (Allerton Park, IL, 1989), Progr. Math., 85, Boston, MA, Birkhäuser Boston, 1990, 451–477 | MR | Zbl

[46] Qiyu Sun, “Refinable functions with compact support”, J. Approx. Theory, 86:2 (1996), 240–252 | DOI | MR | Zbl

[47] A. Tanturri, “Sul numero delle partizioni d'un numero in potenze di 2”, Atti Accad. Naz. Lincei. Rend., 27 (1918), 399–403 | Zbl

[48] R. C. Vaughan, “Bounds for the coefficients of cyclotomic polynomials”, Michigan Math. J., 21:4 (1975), 289–295 | DOI | MR | Zbl

[49] B. L. van der Varden, Algebra, 2-e izd., Nauka, M., 1979, 624 pp. ; B. L. van der Waerden, Algebra, v. I, Heidelberger Taschenbücher, 12, 8. Aufl., Springer-Verlag, Berlin–New York, 1971, ix+272 pp. ; v. II, Heidelberger Taschenbücher, 23, 5. Aufl., 1967, x+300 pp. ; B. L. van der Waerden, Algebra, Reprint. from the English transl. from German of 1970, т. 1, 2, Springer-Verlag, New York, 1991, xiv+265 pp., xii+284 с. | MR | Zbl | MR | Zbl | MR | Zbl | MR | MR | Zbl