On a structure of the fixed point set of homogeneous maps

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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

Abstract
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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.
[2]	M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451.
[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.
[4]	Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577.
[5]	A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974).
[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.
[7]	W. Fulton, "Intersection Theory,", Second edition, 2 (1998). doi: 10.1007/978-1-4612-1700-8.
[8]	Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341.
[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.
[10]	I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974).
[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.
[12]	A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257.
[13]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545.
[14]	J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177. doi: 10.1112/blms/11.2.177.

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.
[2]	M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: II. Applications,, Ann. of Math. (2), 88 (1968), 451.
[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.
[4]	Z. Balanov, Y. Krasnov and A. Kononovich, Projective dynamics of homogeneous systems: Local invariants, syzygies and global residue theorem,, Z, 55 (2012), 577.
[5]	A. Dold, "Lectures on Algebraic Topology,", Berlin, (1974).
[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.
[7]	W. Fulton, "Intersection Theory,", Second edition, 2 (1998). doi: 10.1007/978-1-4612-1700-8.
[8]	Z. V. Kovarik, Similarity and interpolation between projectors,, Acta Sci. Math. (Szeged), 39 (1977), 341.
[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.
[10]	I. R. Shafarevich, "Basic Algebraic Geometry,", Berlin, 213 (1974).
[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.
[12]	A. Tretyakov and H. .Zołądek, A remark about homogeneous polynomial maps,, Topological Methods in Nonlinear Analysis, 19 (2002), 257.
[13]	H. Whitney, Elementary structure of real algebraic varieties,, Ann. Math., 66 (1957), 545.
[14]	J. Zemánek, Idempotents in Banach algebras,, Bull. London Math. Soc., 11 (1979), 177. doi: 10.1112/blms/11.2.177.

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