scholarly journals Gröbner bases for complex Grassmann manifolds

2011 ◽  
Vol 90 (104) ◽  
pp. 23-46 ◽  
Author(s):  
Branislav Prvulovic
Keyword(s):  
Grassmann Manifold ◽  
Polynomial Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Grassmann Manifolds ◽  
Integral Cohomology ◽  
Complex Grassmann Manifold ◽  
Bner Basis

By Borel?s description, integral cohomology of the complex Grassmann manifold Gk,n is a polynomial algebra modulo a well-known ideal. A strong Gr?bner basis for this ideal is obtained when k = 2 and k = 3.

scholarly journals Rational homotopy of maps between certain complex Grassmann manifolds

2018 ◽  
Vol 68 (1) ◽  
pp. 181-196 ◽  
Author(s):  
Prateep Chakraborty ◽  
Shreedevi K. Masuti
Keyword(s):  
Grassmann Manifold ◽  
Dimensional Vector ◽  
Rational Homotopy ◽  
Grassmann Manifolds ◽  
Continuous Map ◽  
Complex Grassmann Manifold ◽  
Vector Subspaces

AbstractLetGn,kdenote the complex Grassmann manifold ofk-dimensional vector subspaces of ℂn. Assumel,k≤ ⌊n/2⌋. We show that, for sufficiently largen, any continuous maph:Gn,l→Gn,kis rationally null homotopic if (i) 1 ≤k<l, (ii) 2 < l <k< 2(l− 1), (iii) 1 < l <k,ldividesnbutldoes not dividek.

Gröbner bases for some flag manifolds and applications

2016 ◽  
Vol 66 (5) ◽  
Author(s):  
Marko Radovanović
Keyword(s):  
Polynomial Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Flag Manifolds ◽  
The Real

AbstractThe mod 2 cohomology of the real flag manifolds is known to be isomorphic to a polynomial algebra modulo a certain ideal. In this paper reduced Gröbner bases for these ideals are obtained in the case of manifolds

scholarly journals Signed polyomino tilings by n-in-line polyominoes and Gröbner bases

2016 ◽  
Vol 99 (113) ◽  
pp. 31-42 ◽  
Author(s):  
Manuela Muzika-Dizdarevic ◽  
Marinko Timotijevic ◽  
Rade Zivaljevic
Keyword(s):  
Homology Group ◽  
Hexagonal Lattice ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Explicit Description ◽  
Division Algorithm ◽  
Triangular Region ◽  
Newton Polynomial ◽  
Bner Bases ◽  
Bner Basis

Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits a signed tiling by three-in-line polyominoes (tribones) if and only if m 2 {9d?1, 9d}d2N. We apply the theory of Gr?bner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m 2 {dn2 ? 1, dn2}d2N. Explicit description of the Gr?bner basis allows us to calculate the ?Gr?bner discrete volume? of a lattice region by applying the division algorithm to its ?Newton polynomial?. Among immediate consequences is a description of the tile homology group for the n-in-line polyomino.

scholarly journals Nonimmersions of complex Grassmann manifolds

1981 ◽  
Vol 24 (1) ◽  
pp. 1-3 ◽  
Author(s):  
Samuel A. Ilori
Keyword(s):  
Normal Bundle ◽  
Grassmann Manifold ◽  
Cohomology Ring ◽  
Euler Class ◽  
Grassmann Manifolds ◽  
Oriented Manifold ◽  
Complex Grassmann Manifold

If an oriented manifold M immerses in codimension k, then the normal bundle has dimension k such that its Euler class χ є Hk(M; Z) and χ2 є H2k(M; Z). (Cf. (3)).If M is the complex Grassmann manifold G2(Cn) of 2-planes in Cn (n = 4, 5,…, 15, 17), then dim M = 4n – 8 ≡ d and we shall show that although M immerses in R2d–1 by classical results (3), M does not immerse in Rd+d/2.The same result was obtained for n = 4 and 5 by Connell (2) and for n = 6 and 7 by the author (6). The nonimmersion results of this paper are new for n = 8, 9, …, 15, 17 and they are an improvement over the result for the general G2(Cn) obtained in (5). In this paper, we use generators of the cohomology ring of G2(Cn) different from those used in (2) and (6) and this simplifies the calculations considerably.

scholarly journals Uniqueness of the Sum of Points of the Period-Five Cycle of Quadratic Polynomials

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Pekka Kosunen
Keyword(s):  
Polynomial Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Quadratic Polynomial ◽  
Extension Theory ◽  
Coordinate Plane ◽  
Quadratic Polynomials

It is well known that the sum of points of the period-five cycle of the quadratic polynomial fc(x)=x2+c is generally not one-valued. In this paper we will show that the sum of cycle points of the curves of period five is at most three-valued on a new coordinate plane and that this result is essentially the best possible. The method of our proof relies on a implementing Gröbner-bases and especially extension theory from the theory of polynomial algebra.

ON GRÖBNER BASES FOR FLAG MANIFOLDS F(1, 1, … , 1, n)

2012 ◽  
Vol 12 (03) ◽  
pp. 1250182
Author(s):  
ZORAN Z. PETROVIĆ ◽  
BRANISLAV I. PRVULOVIĆ
Keyword(s):  
Vector Fields ◽  
Polynomial Algebra ◽  
Gröbner Bases ◽  
Flag Manifold ◽  
Grobner Bases ◽  
Flag Manifolds ◽  
Embedding Dimension ◽  
The Real ◽  
Real Flag Manifold ◽  
Complete Flag

The knowledge of cohomology of a manifold has shown to be quite relevant in various investigations: the question of vector fields, immersion and embedding dimension, and recently even in topological robotics. The method of Gröbner bases is applicable when the cohomology of the manifold is a quotient of a polynomial algebra. The mod 2 cohomology of the real flag manifold F(n1, n2, …, nr) is known to be isomorphic to a polynomial algebra modulo a certain ideal. Reduced Gröbner bases for these ideals are obtained in the case of manifolds F(1, 1, …, 1, n) including the complete flag manifolds (n = 1).

scholarly journals Gröbner bases for operads

2010 ◽  
Vol 153 (2) ◽  
pp. 363-396 ◽  
Author(s):  
Vladimir Dotsenko ◽  
Anton Khoroshkin
Keyword(s):  
Gröbner Bases ◽  
Grobner Bases
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