American Institute of Mathematical Sciences

Advanced
April  2013, 6(4): 1017-1027. doi: 10.3934/dcdss.2013.6.1017

On a structure of the fixed point set of homogeneous maps

Yakov Krasnov 1,2,3, , Alexander Kononovich 1,2,3, and  Grigory Osharovich 1,2,3,

1. 

Department of Mathematics
2. 

Bar-Ilan University
3. 

Ramat-Gan, 52900

Received  June 2011 Revised  September 2011 Published  December 2012

A spectral and inverse spectral problem for homogeneous polynomial maps is discussed.The $m$-independence of vectors based on the symmetric tensor powers performs as a main toolto study the structure of the spectrum. Possible restrictions on this structureare described in terms of syzygies provided by the Euler-Jacobi formula.Applications to projective dynamics are discussed.
Keywords: Fixed points theory, Bzout theorem, Euler-Jacobi formula, multi-linear analysis, syzygies..
Mathematics Subject Classification: Primary: 34A34, 15A69; Secondary: 15A18, 34C14, 34C4.
Citation: Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
References:
[1]

B. Aupetit, Projections in real Banach algebras,, Bull. London Math. Soc., 13 (1981), 412.  doi: 10.1112/blms/13.5.412.  Google Scholar

[2]

M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451.   Google Scholar

[3]

Z. Balanov and Y. Krasnov, Complex structures in real algebras I. Two-dimensional commutative case,, Comm. Algebra, 31 (2003), 4571.  doi: 10.1081/AGB-120022810.  Google Scholar

[4]

Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577.   Google Scholar

[5]

A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974).   Google Scholar

[6]

J. Esterle and J. Giol, Polynomial and polygonal connections between idempotents in finite dimensional real algebras,, Bull. London Math. Soc., 36 (2004), 378.  doi: 10.1112/S0024609303002820.  Google Scholar

[7]

W. Fulton, "Intersection Theory,", Second edition, 2 (1998).  doi: 10.1007/978-1-4612-1700-8.  Google Scholar

[8]

Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341.   Google Scholar

[9]

J. Llibre and V. Pilyugina, Number of invariant straight Lines for homogeneous polynomial vector fields of arbitrary degree and dimension,, J. Dyn. Diff. Equat., 21 (2009), 487.  doi: 10.1007/s10884-009-9141-x.  Google Scholar

[10]

I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974).   Google Scholar

[11]

M. Shub and S. Smale, Complexity of Bézout's theorem. I. Geometric aspects,, J. Amer. Math. Soc., 6 (1993), 459.  doi: 10.2307/2152805.  Google Scholar

[12]

A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257.   Google Scholar

[13]

H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545.   Google Scholar

[14]

J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177.  doi: 10.1112/blms/11.2.177.  Google Scholar

show all references

References:
[1]

B. Aupetit, Projections in real Banach algebras,, Bull. London Math. Soc., 13 (1981), 412.  doi: 10.1112/blms/13.5.412.  Google Scholar

[2]

M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451.   Google Scholar

[3]

Z. Balanov and Y. Krasnov, Complex structures in real algebras I. Two-dimensional commutative case,, Comm. Algebra, 31 (2003), 4571.  doi: 10.1081/AGB-120022810.  Google Scholar

[4]

Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577.   Google Scholar

[5]

A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974).   Google Scholar

[6]

J. Esterle and J. Giol, Polynomial and polygonal connections between idempotents in finite dimensional real algebras,, Bull. London Math. Soc., 36 (2004), 378.  doi: 10.1112/S0024609303002820.  Google Scholar

[7]

W. Fulton, "Intersection Theory,", Second edition, 2 (1998).  doi: 10.1007/978-1-4612-1700-8.  Google Scholar

[8]

Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341.   Google Scholar

[9]

J. Llibre and V. Pilyugina, Number of invariant straight Lines for homogeneous polynomial vector fields of arbitrary degree and dimension,, J. Dyn. Diff. Equat., 21 (2009), 487.  doi: 10.1007/s10884-009-9141-x.  Google Scholar

[10]

I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974).   Google Scholar

[11]

M. Shub and S. Smale, Complexity of Bézout's theorem. I. Geometric aspects,, J. Amer. Math. Soc., 6 (1993), 459.  doi: 10.2307/2152805.  Google Scholar

[12]

A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257.   Google Scholar

[13]

H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545.   Google Scholar

[14]

J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177.  doi: 10.1112/blms/11.2.177.  Google Scholar

[1]

Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291
[2]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, , () : -. doi: 10.3934/era.2021001
[3]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[4]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[5]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
[6]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374
[7]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[8]

Jinsen Zhuang, Yan Zhou, Yonghui Xia. Synchronization analysis of drive-response multi-layer dynamical networks with additive couplings and stochastic perturbations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1607-1629. doi: 10.3934/dcdss.2020279
[9]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[10]

Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021002
[11]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139
[12]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168
[13]

Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605
[14]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395
[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[16]

Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008
[17]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[18]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[19]

Lin Jiang, Song Wang. Robust multi-period and multi-objective portfolio selection. Journal of Industrial & Management Optimization, 2021, 17 (2) : 695-709. doi: 10.3934/jimo.2019130
[20]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences