Totally positive polynomials with small length

• Autorzy: Flammang Valerie , Sac-Épée Jean-Marc
• Języki publikacji: EN

Abstrakty

EN

In this work we provide a list of irreducible monic, totally positive polynomials with integer coefficients and lengths below 2.3679^d for a degree d polynomial. Our approach combines techniques based on auxiliary functions and mixed-integer linear programming algorithms. The list obtained shows that there are polynomials of this type up to degree 16, with the exception of degree 11 where we are very close to the threshold.

Słowa kluczowe

EN

auxiliary functions; length of polynomials

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 3
• Strony: 463--471
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Flammang Valerie

• Institut Élie Cartan de Lorraine, Université de Lorraine-Metz, Ile du Saulcy, 57045 Metz, France

autor

Sac-Épée Jean-Marc

• Institut Élie Cartan de Lorraine, Université de Lorraine-Metz, Ile du Saulcy, 57045 Metz, France

Bibliografia

• Flammang, V. (1994) Mesures de polynômes. Application au diamétre transfini entier. Thése de Doctorat de l’Université de Metz.
• Flammang, V. (2014) On the absolute length of polynomials having all zeros in a sector. J. Number Theory 143, 385–401.
• Flammang, V. (2016) Une nouvelle minoration pour la trace absolue des entiers algébriques totalement positifs. Note, hal-01346165.
• Kronecker, L. (1857) Zwei Sätze über Gleichungen mit ganzzahligen Koeffizienten, J. reine angew. Math. 53, 173–175.
• Langevin, M. (1980) Mesures des polynômmes et des nombres algebriques. Journes de Limoges, provided by G. Rhin.
• Matlab Optimization toolbox. User’s Guide (inline) available at https://www.mathworks.com/help/pdf_doc/optim/optim_tb.pdf
• Mu, Q. and Wu, Q. (2013) The measure of totally positive algebraic integers. J. Number Theory 133, 1, 12–19.
• Otmani, S., Maul, A., Rhin, G., Sac-´Epée, J.-M. (2014) Finding degree 16 monic irreducible integer polynomials of minimal trace by optimization methods. Exp. Math. 23, 1, 1–5.
• Pari/GP (2016) User’s Guide to the Pari library, available at https://pari.math.u-bordeaux.fr/pub/pari/manuals/2.7.6/libpari.pdf
• Rustem, B. (2000) Integer Linear Programming, available at https://www.doc.ic.ac.uk/~br/berc/integerprog.pdf
• Smyth, C. J. (1984) The mean value of totally real algebraic numbers. Math. Comp. 42, 663–681.
• Smyth, C.J. (2008) The Mahler measure of algebraic numbers: a survey. Number Theory and Polynomials, London Mathematical Society Lecture Note Series 352, Cambridge University Press, Cambridge, 322–349.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).