Publication: Semialgebraic sets and real binary forms decompositions
Loading...
Official URL
Full text at PDC
Publication Date
2021
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Citations
Exportar
Abstract
The Waring Problem over polynomial rings asks how to decompose a homogeneous polynomial p of degree d as a linear combination of d-th powers of linear forms. In this work we give an algorithm to obtain a real Waring decomposition of any given real binary form p of length at most its degree. In fact, we construct a semialgebraic family of Waring decompositions for p. We illustrate our results with some examples.
Description
UCM subjects
Unesco subjects
Keywords
Citation
Alexander, J., Hirschowitz, A., 1995. Polynomial interpolation in several variables. J. Algebraic Geom. 4, 201–222.
Ballico, E., Bernardi, A., 2013. Real and complex rank for real symmetric tensors with low ranks. Algebra 2013, 794054.
Basu, S., Pollack, R., Roy, M.F., 2006. Algorithms in Real Algebraic Geometry, vol. 10. Springer-Verlag Berlin Heidelberg. https://perso .univ-rennes1.fr /marie -francoise .roy /bpr-ed2 -posted3 .pdf.
Bender, M.R., Faugère, J.C., Perret, L., Tsigaridas, E., 2016. A superfast randomized algorithm to decompose binary forms. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, NY, USA, pp. 79–86.
Bender, M.R., Faugère, J.C., Perret, L., Tsigaridas, E., 2020. A nearly optimal algorithm to decompose binary forms. J. Symbolic. Comput. https://doi .org /10 .1016 /j .jsc .2020 .06 .002.
Blekherman, G., 2015. Typical real ranks of binary forms. Found. Comput. Math. 15, 793–798.
https://www.springerprofessional .de /en /typical -real -ranks -of -binary-forms /11689832.
Bochnak, J., Coste, M., Roy, M.F., 1998. Real Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 36. Springer-Verlag, Berlin.
Boij, M., Carlini, E., Geramita, A., 2011. Monomials as sums of powers: the real binary case. Proc. Am. Math. Soc. 139, 3039–3043. https://www.ams .org /journals /proc /2011 -139 -09 /S0002 -9939 -2011 -11018 -9 /S0002 -9939 -2011 -11018 -9 .pdf.
Brachat, J., Comon, P., Mourrain, B., Tsigaridas, E., 2010. Symmetric tensor decomposition. Linear Algebra Appl. 433, 1851–1872.
Brustenga i Moncusí, L., Masuti, S.K., 2019. On the Waring rank of binary forms: the binomial formula and a dihedral cover of rank two forms. arXiv:1901.08320.
Carlini, E., Catalisano, M., Geramita, A.V., 2012. The solution to the Waring problem for monomials and the sum of coprime monomials. J. Algebra 370, 5–14.
Causa, A., Re, R., 2011. On the maximum rank of a real binary form. Ann. Mat. Pura Appl. 190, 55–59.
Comon, P., Lim, L.H., Qi, Y., Ye, K., 2020. Topology of tensor ranks. Adv. Math. 367, 107128.
Comon, P., Mourrain, B., 1996. Decomposition of quantics in sums of powers of linear forms. Signal Process. 53, 93–107.
Comon, P., Ottaviani, G., 2012. On the typical rank of real binary forms. Linear Multilinear Algebra 60, 657-667. Fröberg, R., Lundqvist, S., Oneto, A., Shapiro, B., 2018. Algebraic stories from one and from the other pockets. Arnold Math. J. 4, 137–160. https://arxiv.org /pdf /1801.01692v1.pdf.
García-Marco, I., Koiran, P., Pecatte, T., 2017. Reconstruction algorithms for sums of affine powers. In: Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation. ACM, New York, NY, USA, pp. 317–324.Helmke, U., 1992. Waring’s problem for binary forms. J. Pure Appl. Algebra 80, 29–45. https://doi .org /10 .1016 /0022 -4049(92 )90067 -P.
Li, L., 2000. On the arithmetic operational complexity for solving Vandermonde linear equations. Jpn. J. Ind. Appl. Math. 17, 15–18.
Łojasiewicz, S., 1991. Introduction to Complex Analytic Geometry. Ed. Birkhäuser.
Pan, V., Sadikou, A., Landowne, E., Tiga, O., 1993. A new approach to fast polynomial interpolation and multipoint evaluation. Comput. Math. Appl. 25, 25–30.
Pratt, K., 2018. Faster algorithms via Waring decompositions. In: The Proceedings of FOCS.
Reznick, B., 2010. Laws of inertia in higher degree binary forms. Proc. Am. Math. Soc. 138, 815–826.
Tokcan, N., 2017. On the Waring rank of binary forms. Linear Algebra Appl. 524, 250–262.220