### D.5.2 chern_lib

Library:
  chern.lib
Purpose:
  Symbolic Computations with Chern classes, Computation of Chern classes

Author:
  Oleksandr Iena, o.g.yena@gmail.com

Overview:
  A toolbox for symbolic computations with Chern classes. The Aluffi's algorithms for computation of characteristic classes of algebraic varieties (Segre, Fulton, Chern-Schwartz-MacPherson classes) are implemented as well.

References:
  [1] Aluffi, Paolo Computing characteristic classes of projective schemes. Journal of Symbolic Computation, 35 (2003), 3-19. [2] Iena, Oleksandr, On symbolic computations with Chern classes: remarks on the library chern.lib for Singular,
  http://hdl.handle.net/10993/22395, 2015.
  [3] Lascoux, Alain, Classes de Chern d'un produit tensoriel. C. R. Acad. Sci., Paris, Ser. A 286, 385-387 (1978). [4] Manivel, Laurent Chern classes of tensor products, arXiv 1012.0014, 2010.

Procedures:

> D.5.2.1 symm |  | symmetric functions in the entries of l
> D.5.2.2 symNsym |  | symmetric and non-symmetric parts of a polynomial f
> D.5.2.3 CompleteHomog |  | complete homogeneous symmetric functions
> D.5.2.4 segre |  | Segre classes in terms of Chern classes
> D.5.2.5 chern |  | Chern classes in terms of Segre classes
> D.5.2.6 chNum |  | the non-zero Chern numbers in degree N in the entries of c
> D.5.2.7 chNumbers |  | the Chern numbers in degree N in the entries of c
> D.5.2.8 sum_of_powers |  | the sum of k-th powers of the entries of l
> D.5.2.9 powSumSym |  | the sums of powers [up to degree N] in terms of the elementary symmetric polynomials (entries of l)
> D.5.2.10 chAll |  | Chern character in terms of the Chern classes
> D.5.2.11 chAllInv |  | Chern classes in terms of the Chern character
> D.5.2.12 chHE |  | the highest term of the Chern character
> D.5.2.13 ChernRootsSum |  | the Chern roots of a direct sum
> D.5.2.14 chSum |  | the Chern classes of a direct sum
> D.5.2.15 ChernRootsDual |  | the Chern roots of the dual vector bundle
> D.5.2.16 chDual |  | the Chern classes of the dual vector bundle
> D.5.2.17 ChernRootsProd |  | the Chern roots of a tensor product of vector bundles
> D.5.2.18 chProd |  | Chern classes of a tensor product of vector bundles
> D.5.2.19 chProdE |  | Chern classes of a tensor product of vector bundles
> D.5.2.20 chProdL |  | Chern classes of a tensor product of vector bundles
> D.5.2.21 chProdLP |  | total Chern class of a tensor product of vector bundles
> D.5.2.22 chProdM |  | Chern classes of a tensor product of vector bundles
> D.5.2.23 chProdMP |  | total Chern class of a tensor product of vector bundles
> D.5.2.24 ChernRootsHom |  | the Chern roots of a Hom vector bundle
> D.5.2.25 chHom |  | Chern classes of the Hom-vector bundle
> D.5.2.26 ChernRootsSymm |  | the Chern roots of the n-th symmetric power of a vector bundle with Chern roots from l
> D.5.2.27 ChernRootsWedge |  | the Chern roots of the n-th exterior power of a vector bundle with Chern roots from l
> D.5.2.28 chSymm |  | the rank and the Chern classes of the k-th symmetric power of a vector bundle of rank r with Chern classes c
> D.5.2.29 chSymm2L |  | the rank and the Chern classes of the second symmetric power of a vector bundle of rank r with Chern classes c
> D.5.2.30 chSymm2LP |  | the total Chern class of the second symmetric power of a vector bundle of rank r with Chern classes c
> D.5.2.31 chWedge |  | the rank and the Chern classes of the k-th exterior power of a vector bundle of rank r with Chern classes c
> D.5.2.32 chWedge2L |  | the rank and the Chern classes of the second exterior power of a vector bundle of rank r with Chern classes c
> D.5.2.33 chWedge2LP |  | the total Chern class of the second exterior power of a vector bundle of rank r with Chern classes c
> D.5.2.34 todd |  | the Todd class
> D.5.2.35 toddE |  | the highest term of the Todd class
> D.5.2.36 Bern |  | the second Bernoulli numbers
> D.5.2.37 tdCf |  | the coefficients of the Todd class of a line bundle
> D.5.2.38 tdTerms |  | the terms of the Todd class of a line bundle corresponding to the Chern root t
> D.5.2.39 tdFactor |  | the Todd class of a line bundle corresponding to the Chern root t
> D.5.2.40 cProj |  | the total Chern class of (the tangent bundle on) the projective space P_n
> D.5.2.41 chProj |  | the Chern character of (the tangent bundle on) the projective space P_n
> D.5.2.42 tdProj |  | the Todd class of (the tangent bundle on) the projective space P_n
> D.5.2.43 eulerChProj |  | Euler characteristic of a vector bundle on the projective space P_n via Hirzebruch-Riemann-Roch theorem
> D.5.2.44 chNumbersProj |  | the Chern numbers of the projective space P_n
> D.5.2.45 classpoly |  | polynomial in t with coefficients from l (without constant term)
> D.5.2.46 chernPoly |  | Chern polynomial (constant term 1)
> D.5.2.47 chernCharPoly |  | polynomial in t corresponding to the Chern character (constant term r)
> D.5.2.48 toddPoly |  | polynomial in t corresponding to the Todd class (constant term 1)
> D.5.2.49 rHRR |  | the main ingredient of the right-hand side of the Hirzebruch-Riemann-Roch formula
> D.5.2.50 SchurS |  | the Schur polynomial corresponding to partition I in terms of the Segre classes S
> D.5.2.51 SchurCh |  | the Schur polynomial corresponding to partition I in terms of the Chern classes C
> D.5.2.52 part |  | partitions of integers not exceeding n into m non-negative summands
> D.5.2.53 dualPart |  | partition dual to I
> D.5.2.54 PartC |  | the complement of a partition with respect to m
> D.5.2.55 partOver |  | partitions over a given partition J with summands not exceeding n
> D.5.2.56 partUnder |  | partitions under a given partition J
> D.5.2.57 SegreA |  | Segre class of the projective subscheme defined by I
> D.5.2.58 FultonA |  | Fulton class of the projective subscheme defined by I
> D.5.2.59 CSMA |  | Chern-Schwartz-MacPherson class of the projective subscheme defined by I
> D.5.2.60 EulerAff |  | Euler characteristic of the affine subvariety defined by I
> D.5.2.61 EulerProj |  | Euler characteristic of the projective subvariety defined by I

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