# American Institute of Mathematical Sciences

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

## On a structure of the fixed point set of homogeneous maps

 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.
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.

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##### 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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