Thom polynomials of Morin singularities

Abstract

We prove a formula for Thom polynomials of $A_d$ singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to $d=6$.

  • [arnold] V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasilliev, "Singularity theory. I," in Dynamical Systems. VI, New York: Springer-Verlag, 1993, vol. 6, p. iv.
    @incollection{arnold, MRKEY = {1230637},
      AUTHOR={Arnold, V. I. and Goryunov, V. V. and Lyashko, O. V. and Vasilliev, V. A.},
      TITLE = {Singularity theory. {I}},
      BOOKTITLE={Dynamical Systems. {VI}},
      SERIES = {Encycl. Math. Sci.},
      VOLUME = {6},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1993},
      PAGES = {iv+245},
      ISBN = {3-540-50583-0},
      MRCLASS = {58C27 (32Sxx)},
      MRNUMBER = {1230637},
      ZBLNUMBER = {0786.00006},
      }
  • [berczi] G. Bérczi, Multidegrees of singularities and nonreductive quotients, 2008.
    @misc{berczi,
      author={B{é}rczi, G.},
      TITLE={Multidegrees of singularities and nonreductive quotients},
      NOTE={Ph.D. thesis, BME, Budapest},
      YEAR={2008},
     }
  • [bfr] G. Bérczi, L. M. Fehér, and R. Rimányi, "Expressions for resultants coming from the global theory of singularities," in Topics in Algebraic and Noncommutative Geometry, Providence, RI: Amer. Math. Soc., 2003, vol. 324, pp. 63-69.
    @incollection {bfr, MRKEY = {1986113},
      AUTHOR = {B{é}rczi, G. and Feh{é}r, L. M. and Rim{á}nyi, R.},
      TITLE = {Expressions for resultants coming from the global theory of singularities},
      BOOKTITLE = {Topics in Algebraic and Noncommutative Geometry},
      VENUE={{L}uminy/{A}nnapolis, {MD},
      2001},
      SERIES = {Contemp. Math.},
      VOLUME = {324},
      PAGES = {63--69},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2003},
      MRCLASS = {58K30 (14B05 57R45)},
      MRNUMBER = {1986113},
      MRREVIEWER = {Guangfeng Jiang},
      ZBLNUMBER = {1052.57043},
      }
  • [bgv] N. Berline, E. Getzler, and M. Vergne, Heat kernels and Dirac operators, New York: Springer-Verlag, 1992.
    @book {bgv, MRKEY = {1215720},
      AUTHOR = {Berline, Nicole and Getzler, Ezra and Vergne, Mich{è}le},
      TITLE = {Heat kernels and {D}irac operators},
      SERIES = {Grundl. Math. Wissen.},
      NUMBER = {298},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1992},
      PAGES = {viii+369},
      ISBN = {3-540-53340-0},
      MRCLASS = {58G10 (58G11)},
      MRNUMBER = {1215720},
      MRREVIEWER = {Alejandro Uribe},
      ZBLNUMBER = {0744.58001},
      }
  • [bv] Go to document N. Berline and M. Vergne, "Zéros d’un champ de vecteurs et classes caractéristiques équivariantes," Duke Math. J., vol. 50, iss. 2, pp. 539-549, 1983.
    @article {bv, MRKEY = {0705039},
      AUTHOR = {Berline, Nicole and Vergne, Mich{è}le},
      TITLE = {Zéros d'un champ de vecteurs et classes caractéristiques équivariantes},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {50},
      YEAR = {1983},
      NUMBER = {2},
      PAGES = {539--549},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {58G10 (57R20 58A10)},
      MRNUMBER = {0705039},
      MRREVIEWER = {Italo Jos{é} Dejter},
      DOI = {10.1215/S0012-7094-83-05024-X},
      }
  • [boardman] J. M. Boardman, "Singularities of differentiable maps," Inst. Hautes Études Sci. Publ. Math., iss. 33, pp. 21-57, 1967.
    @article {boardman, MRKEY = {0231390},
      AUTHOR = {Boardman, J. M.},
      TITLE = {Singularities of differentiable maps},
      JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
      FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
      NUMBER = {33},
      YEAR = {1967},
      PAGES = {21--57},
      ISSN = {0073-8301},
      MRCLASS = {57.20},
      MRNUMBER = {0231390},
      MRREVIEWER = {I. R. Porteous},
      ZBLNUMBER = {0165.56803},
      }
  • [bott] Go to document R. Bott, "Vector fields and characteristic numbers," Michigan Math. J., vol. 14, pp. 231-244, 1967.
    @article {bott, MRKEY = {0211416},
      AUTHOR = {Bott, Raoul},
      TITLE = {Vector fields and characteristic numbers},
      JOURNAL = {Michigan Math. J.},
      FJOURNAL = {The Michigan Mathematical Journal},
      VOLUME = {14},
      YEAR = {1967},
      PAGES = {231--244},
      ISSN = {0026-2285},
      MRCLASS = {57.34 (53.75)},
      MRNUMBER = {0211416},
      MRREVIEWER = {M. F. Atiyah},
      ZBLNUMBER = {0145.43801},
      DOI = {10.1307/mmj/1028999721},
     }
  • [botttu] R. Bott and L. W. Tu, Differential forms in algebraic topology, New York: Springer-Verlag, 1982, vol. 82.
    @book {botttu, MRKEY = {0658304},
      AUTHOR = {Bott, Raoul and Tu, Loring W.},
      TITLE = {Differential forms in algebraic topology},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {82},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1982},
      PAGES = {xiv+331},
      ISBN = {0-387-90613-4},
      MRCLASS = {57R19 (55-02 58-01 58A12)},
      MRNUMBER = {0658304},
      MRREVIEWER = {Hansklaus Rummler},
      ZBLNUMBER = {0496.55001},
      }
  • [damon] J. Damon, Thom polynomials for contact class singularities, 1972.
    @misc{damon,
      author={Damon, J.},
      TITLE={Thom polynomials for contact class singularities},
      NOTE={Ph.D. thesis, Harvard University},
      YEAR={1972},
     }
  • [kirwan] B. Doran and F. Kirwan, Towards nonreductive geometric invariant theory.
    @misc{kirwan,
      author={Doran, B. and Kirwan, F.},
      TITLE={Towards nonreductive geometric invariant theory},
      NOTE={to appear},
     }
  • [dv] M. Duflo and M. Vergne, "Orbites coadjointes et cohomologie équivariante," in The Orbit Method in Representation Theory, Boston, MA: Birkhäuser, 1990, vol. 82, pp. 11-60.
    @incollection {dv, MRKEY = {1095340},
      AUTHOR = {Duflo, Michel and Vergne, Mich{è}le},
      TITLE = {Orbites coadjointes et cohomologie équivariante},
      BOOKTITLE = {The Orbit Method in Representation Theory},
      VENUE={{C}openhagen, 1988},
      SERIES = {Progr. Math.},
      VOLUME = {82},
      PAGES = {11--60},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1990},
      MRCLASS = {22E30 (22E60)},
      MRNUMBER = {1095340},
      MRREVIEWER = {Brian E. Blank},
      ZBLNUMBER = {0741.22008},
      }
  • [eisenbudharris] D. Eisenbud and J. Harris, The Geometry of Schemes, New York: Springer-Verlag, 2000, vol. 197.
    @book {eisenbudharris, MRKEY = {1730819},
      AUTHOR = {Eisenbud, David and Harris, Joe},
      TITLE = {The Geometry of Schemes},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {197},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2000},
      PAGES = {x+294},
      ISBN = {0-387-98638-3; 0-387-98637-5},
      MRCLASS = {14A15 (14-01 14C05 14H50 14N05)},
      MRNUMBER = {1730819},
      MRREVIEWER = {Allen B. Altman},
      ZBLNUMBER = {0960.14002},
      }
  • [eisenbud] D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry, New York: Springer-Verlag, 1995, vol. 150.
    @book {eisenbud, MRKEY = {1322960},
      AUTHOR = {Eisenbud, David},
      TITLE = {Commutative Algebra, with a View Toward Algebraic Geometry},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {150},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {xvi+785},
      ISBN = {0-387-94268-8; 0-387-94269-6},
      MRCLASS = {13-01 (14A05)},
      MRNUMBER = {1322960},
      MRREVIEWER = {Matthew Miller},
      ZBLNUMBER = {0819.13001},
      }
  • [rimfeher] L. M. Fehér and R. Rimányi, "Thom polynomial computing strategies. A survey," in Singularity theory and its applications, Tokyo: Math. Soc. Japan, 2006, pp. 45-53.
    @incollection {rimfeher, MRKEY = {2313407},
      AUTHOR = {Feh{é}r, L{á}szl{ó} M. and Rim{á}nyi, Rich{á}rd},
      TITLE = {Thom polynomial computing strategies. {A} survey},
      BOOKTITLE = {Singularity theory and its applications},
      SERIES = {Adv. Stud. Pure Math.},
      NUMBER = {43},
      PAGES = {45--53},
      PUBLISHER = {Math. Soc. Japan},
      ADDRESS = {Tokyo},
      YEAR = {2006},
      MRCLASS = {58K65 (55N91 57R45)},
      MRNUMBER = {2008m:58084},
      MRREVIEWER = {J. M. Boardman},
      ZBLNUMBER = {1135.58303},
      }
  • [rimanyi2] L. M. Fehér and R. Rimányi, "Calculation of Thom polynomials and other cohomological obstructions for group actions," in Real and Complex Singularities, Providence, RI: Amer. Math. Soc., 2004, vol. 354, pp. 69-93.
    @incollection {rimanyi2, MRKEY = {2087805},
      AUTHOR = {Feh{é}r, L{á}szl{ó} M. and Rim{á}nyi, Rich{á}rd},
      TITLE = {Calculation of {T}hom polynomials and other cohomological obstructions for group actions},
      BOOKTITLE = {Real and Complex Singularities},
      SERIES = {Contemp. Math.},
      VOLUME = {354},
      PAGES = {69--93},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      MRCLASS = {58K30 (32S20 57R45)},
      MRNUMBER = {2087805},
      MRREVIEWER = {Harry Tamvakis},
      ZBLNUMBER = {1074.32008},
      }
  • [fr] Go to document L. M. Fehér and R. Rimányi, "On the structure of Thom polynomials of singularities," Bull. Lond. Math. Soc., vol. 39, iss. 4, pp. 541-549, 2007.
    @article {fr, MRKEY = {2346933},
      AUTHOR = {Feh{é}r, L. M. and Rim{á}nyi, R.},
      TITLE = {On the structure of {T}hom polynomials of singularities},
      JOURNAL = {Bull. Lond. Math. Soc.},
      FJOURNAL = {Bulletin of the London Mathematical Society},
      VOLUME = {39},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {541--549},
      ISSN = {0024-6093},
      MRCLASS = {32S20 (58K65)},
      MRNUMBER = {2346933},
      MRREVIEWER = {Jan Stevens},
      DOI = {10.1112/blms/bdm023},
      ZBLNUMBER = {1140.32019},
      }
  • [fulton] W. Fulton, Intersection Theory, New York: Springer-Verlag, 1984, vol. 2.
    @book {fulton, MRKEY = {0732620},
      AUTHOR = {Fulton, William},
      TITLE = {Intersection Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {2},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1984},
      PAGES = {xi+470},
      ISBN = {3-540-12176-5},
      MRCLASS = {14C17 (14-02 14C40)},
      MRNUMBER = {0732620},
      MRREVIEWER = {Werner Kleinert},
      ZBLNUMBER = {0541.14005},
      }
  • [toric] W. Fulton, Introduction to Toric Varieties, Princeton, NJ: Princeton Univ. Press, 1993, vol. 131.
    @book {toric, MRKEY = {1234037},
      AUTHOR = {Fulton, William},
      TITLE = {Introduction to Toric Varieties},
      SERIES = {Ann. of Math. Stud.},
      VOLUME = {131},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1993},
      PAGES = {xii+157},
      ISBN = {0-691-00049-2},
      MRCLASS = {14M25 (14-02 14J30)},
      MRNUMBER = {1234037},
      MRREVIEWER = {T. Oda},
      ZBLNUMBER = {0813.14039},
      }
  • [tableaux] W. Fulton, Young Tableaux, Cambridge: Cambridge Univ. Press, 1997, vol. 35.
    @book {tableaux, MRKEY = {1464693},
      AUTHOR = {Fulton, William},
      TITLE = {Young Tableaux},
      SERIES = {London Math. Soc. Stud. Texts},
      VOLUME = {35},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      PAGES = {x+260},
      ISBN = {0-521-56144-2; 0-521-56724-6},
      MRCLASS = {05E10 (05E05 05E15 14M15 20G05)},
      MRNUMBER = {1464693},
      MRREVIEWER = {Tadeusz J{ó}zefiak},
      ZBLNUMBER = {0878.14034},
      }
  • [gaffney] T. Gaffney, "The Thom polynomial of $\overline{\sum ^{1111}}$," in Singularities, Part 1, Providence, R.I.: Amer. Math. Soc., 1983, vol. 40, pp. 399-408.
    @incollection {gaffney, MRKEY = {0713080},
      AUTHOR = {Gaffney, Terence},
      TITLE = {The {T}hom polynomial of {$\overline{\sum \sp{1111}}$}},
      BOOKTITLE = {Singularities, {P}art 1},
      VENUE={{A}rcata, {C}alif., 1981},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {40},
      PAGES = {399--408},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1983},
      MRCLASS = {58C27 (57R45)},
      MRNUMBER = {0713080},
      MRREVIEWER = {Felice Ronga},
      ZBLNUMBER = {0531.58012},
      }
  • [gasrei] Go to document V. Gasharov and V. Reiner, "Cohomology of smooth Schubert varieties in partial flag manifolds," J. London Math. Soc., vol. 66, iss. 3, pp. 550-562, 2002.
    @article {gasrei, MRKEY = {1934291},
      AUTHOR = {Gasharov, V. and Reiner, V.},
      TITLE = {Cohomology of smooth {S}chubert varieties in partial flag manifolds},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {66},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {550--562},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {14M15 (05E15 14F25)},
      MRNUMBER = {1934291},
      MRREVIEWER = {Raika Dehy},
      DOI = {10.1112/S0024610702003605},
      ZBLNUMBER = {1064.14056},
      }
  • [gs] V. W. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, New York: Springer-Verlag, 1999.
    @book {gs, MRKEY = {1689252},
      AUTHOR = {Guillemin, Victor W. and Sternberg, Shlomo},
      TITLE = {Supersymmetry and Equivariant de {R}ham Theory},
      SERIES = {Mathematics Past and Present},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      PAGES = {xxiv+228},
      ISBN = {3-540-64797-X},
      MRCLASS = {53D20 (55N91 55R20 55R40 57R91 57T15 58C50 58D30)},
      MRNUMBER = {1689252},
      MRREVIEWER = {Liviu I. Nicolaescu},
      ZBLNUMBER = {0934.55007},
      }
  • [griffiths] P. Griffiths and J. Harris, Principles of Algebraic Geometry, New York: Wiley-Interscience [John Wiley & Sons], 1978.
    @book {griffiths, MRKEY = {0507725},
      AUTHOR = {Griffiths, Phillip and Harris, Joseph},
      TITLE = {Principles of Algebraic Geometry},
      SERIES = {Pure Appl. Math.},
      PUBLISHER = {Wiley-Interscience [John Wiley \& Sons]},
      ADDRESS = {New York},
      YEAR = {1978},
      PAGES = {xii+813},
      ISBN = {0-471-32792-1},
      MRCLASS = {14-01},
      MRNUMBER = {0507725},
      MRREVIEWER = {Gerhard Pfister},
      ZBLNUMBER = {0408.14001},
      }
  • [haef] A. Haefliger and A. Kosiński, "Un théorème de Thom sur les singularités des applications différentiables," Séminaire Henri Cartan, vol. 9, 1956-57.
    @article{haef,
      author={Haefliger, A. and Kosiński, A.},
      TITLE={Un théorème de {T}hom sur les singularités des applications différentiables},
      JOURNAL={Séminaire Henri Cartan},
      VOLUME={9},
      YEAR={1956-57},
      NOTE={Secrétariat math., Paris, 1958},
     }
  • [hartshorne] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
    @book {hartshorne, MRKEY = {0463157},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Algebraic Geometry},
      SERIES = {Grad. Texts in Math.},
      VOLUME={52},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1977},
      PAGES = {xvi+496},
      ISBN = {0-387-90244-9},
      MRCLASS = {14-01},
      MRNUMBER = {{0463157}},
      MRREVIEWER = {Robert Speiser},
      ZBLNUMBER = {0367.14001},
      }
  • [joseph] Go to document A. Joseph, "On the variety of a highest weight module," J. Algebra, vol. 88, iss. 1, pp. 238-278, 1984.
    @article {joseph, MRKEY = {0741942},
      AUTHOR = {Joseph, Anthony},
      TITLE = {On the variety of a highest weight module},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {88},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {238--278},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {17B35 (17B10)},
      MRNUMBER = {0741942},
      MRREVIEWER = {James E. Humphreys},
      DOI = {10.1016/0021-8693(84)90100-5},
      ZBLNUMBER = {0539.17006},
      }
  • [kazarian] Go to document M. Kazarian, Thom polynomials.
    @misc{kazarian,
      author={Kazarian, M.},
      TITLE={Thom polynomials},
      NOTE={Lecture notes of talks given at the {S}ingularity {T}heory {C}onference, {S}apporo, 2003},
      URL={http://www.mi.ras.ru/~kazarian/#publ},
     }
  • [komuves] B. KHomHuves, Thom polynomials via restriction equations, 2003.
    @misc{komuves,
      author={Kőműves, B.},
      TITLE={Thom polynomials via restriction equations},
      NOTE={Ms.C. thesis, ELTE},
      YEAR={2003},
     }
  • [laksan] Go to document V. Lakshmibai and B. Sandhya, "Criterion for smoothness of Schubert varieties in ${ Sl}(n)/B$," Proc. Indian Acad. Sci. Math. Sci., vol. 100, iss. 1, pp. 45-52, 1990.
    @article {laksan, MRKEY = {1051089},
      AUTHOR = {Lakshmibai, V. and Sandhya, B.},
      TITLE = {Criterion for smoothness of {S}chubert varieties in {${\rm Sl}(n)/B$}},
      JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.},
      FJOURNAL = {Indian Academy of Sciences. Proceedings. Mathematical Sciences},
      VOLUME = {100},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {45--52},
      ISSN = {0253-4142},
      MRCLASS = {14M15 (14L35)},
      MRNUMBER = {1051089},
      MRREVIEWER = {H. H. Andersen},
      DOI = {10.1007/BF02881113},
      ZBLNUMBER = {0714.14033},
      }
  • [lee] J. M. Lee, Introduction to Smooth Manifolds, New York: Springer-Verlag, 2003, vol. 218.
    @book {lee, MRKEY = {1930091},
      AUTHOR = {Lee, John M.},
      TITLE = {Introduction to Smooth Manifolds},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {218},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2003},
      PAGES = {xviii+628},
      ISBN = {0-387-95495-3},
      MRCLASS = {58-01 (53-01 57-01)},
      MRNUMBER = {1930091},
      MRREVIEWER = {J{ü}rgen Eichhorn},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {1030.53001},
      }
  • [martinet] Go to document J. Martinet, "Déploiements versels des applications différentiables et classification des applications stables," in Singularités d’Applications Différentiables, New York: Springer-Verlag, 1976, vol. 535, pp. 1-44.
    @incollection {martinet, MRKEY = {0649264},
      AUTHOR = {Martinet, Jean},
      TITLE = {Déploiements versels des applications différentiables et classification des applications stables},
      BOOKTITLE = {Singularités d'Applications Différentiables},
      VENUE={{S}ém., {P}lans-sur-{B}ex, 1975},
      PAGES = {1--44},
      SERIES={Lecture Notes in Math.},
      VOLUME={535},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1976},
      MRCLASS = {58C25},
      MRNUMBER = {0649264},
      MRREVIEWER = {F. Berguier},
      ZBLNUMBER = {0362.58004},
      DOI = {10.1007/BFb0080494},
     }
  • [mq] Go to document V. Mathai and D. Quillen, "Superconnections, Thom classes, and equivariant differential forms," Topology, vol. 25, iss. 1, pp. 85-110, 1986.
    @article {mq, MRKEY = {0836726},
      AUTHOR = {Mathai, Varghese and Quillen, Daniel},
      TITLE = {Superconnections, {T}hom classes, and equivariant differential forms},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {25},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {85--110},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {58A10 (32L10 53C05 57R20 58G99)},
      MRNUMBER = {0836726},
      MRREVIEWER = {Thomas Schmitt},
      DOI = {10.1016/0040-9383(86)90007-8},
      ZBLNUMBER = {0592.55015},
      }
  • [mather] J. N. Mather, "On Thom-Boardman singularities," in Dynamical Systems, New York: Academic Press, 1973, pp. 233-248.
    @incollection {mather, MRKEY = {0353359},
      AUTHOR = {Mather, John N.},
      TITLE = {On {T}hom-{B}oardman singularities},
      BOOKTITLE = {Dynamical Systems},
      VENUE={{P}roc. {S}ympos., {U}niv. {B}ahia, {S}alvador, 1971},
      PAGES = {233--248},
      PUBLISHER = {Academic Press},
      ADDRESS = {New York},
      YEAR = {1973},
      MRCLASS = {58C25},
      MRNUMBER = {0353359},
      MRREVIEWER = {Bernard Teissier},
      ZBLNUMBER = {0292.58004},
      }
  • [mather2] J. N. Mather, "Stability of $C^{\infty }$ mappings. VI: The nice dimensions," in Proceedings of Liverpool Singularities-Symposium, I (1969/70), New York, 1971, pp. 207-253.
    @inproceedings {mather2, MRKEY = {0293670},
      AUTHOR = {Mather, John N.},
      TITLE = {Stability of {$C\sp{\infty }$} mappings. {VI}: {T}he nice dimensions},
      BOOKTITLE = {Proceedings of {L}iverpool {S}ingularities-{S}ymposium, {I} (1969/70)},
      PAGES = {207--253},
      SERIES={Lecture Notes in Math.},
      VOLUME={192},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      MRCLASS = {58F10},
      MRNUMBER = {0293670},
      MRREVIEWER = {I. R. Porteous},
      ZBLNUMBER = {0211.56105},
     }
  • [morin] B. Morin, "Formes canoniques des singularités d’une application différentiable," C.R. Acad. Sci. Paris, vol. 260, pp. 5662-5665; 6503, 1965.
    @article{morin,
      author={Morin, B.},
      TITLE={ Formes canoniques des singularités d'une application différentiable},
      JOURNAL={C.R. Acad. Sci. Paris},
      VOLUME={260},
      YEAR={1965},
      PAGES={5662--5665; 6503--6506},
      ZBLNUMBER = {0178.26801},
      }
  • [mumford] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Third ed., New York: Springer-Verlag, 1994, vol. 34.
    @book {mumford, MRKEY = {1304906},
      AUTHOR = {Mumford, D. and Fogarty, J. and Kirwan, F.},
      TITLE = {Geometric Invariant Theory},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {34},
      EDITION = {Third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1994},
      PAGES = {xiv+292},
      ISBN = {3-540-56963-4},
      MRCLASS = {14D25 (58E05 58F05)},
      MRNUMBER = {1304906},
      MRREVIEWER = {Yi Hu},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0797.14004},
      ZBLNUMBER = {0504.14008},
      ZBLNUMBER = {0147.39304},
      }
  • [milsturm] E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, New York: Springer-Verlag, 2005, vol. 227.
    @book {milsturm, MRKEY = {2110098},
      AUTHOR = {Miller, Ezra and Sturmfels, Bernd},
      TITLE = {Combinatorial Commutative Algebra},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {227},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {xiv+417},
      ISBN = {0-387-22356-8},
      MRCLASS = {13-01 (05-01 05E99 13D02 14M15 14M25)},
      MRNUMBER = {2110098},
      MRREVIEWER = {Joseph Gubeladze},
      ZBLNUMBER = {1090.13001},
      }
  • [newstead] P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Bombay: Tata Institute of Fundamental Research, 1978, vol. 51.
    @book {newstead, MRKEY = {0546290},
      AUTHOR = {Newstead, P. E.},
      TITLE = {Introduction to Moduli Problems and Orbit Spaces},
      SERIES = {Tata Inst. Fund. Res. Lect. Math. Phys.},
      VOLUME = {51},
      PUBLISHER = {Tata Institute of Fundamental Research},
      ADDRESS = {Bombay},
      YEAR = {1978},
      PAGES = {vi+183},
      ISBN = {0-387-08851-2},
      MRCLASS = {14-02 (14D20)},
      MRNUMBER = {0546290},
      MRREVIEWER = {G. Horrocks},
      ZBLNUMBER = {0411.14003},
      }
  • [forporteous] Go to document I. R. Porteous, "Simple singularities of maps," in Proceedings of Liverpool Singularities Symposium, I (1969/70), New York, 1971, pp. 286-307.
    @inproceedings {forporteous, MRKEY = {0293646},
      AUTHOR = {Porteous, Ian R.},
      TITLE = {Simple singularities of maps},
      BOOKTITLE = {Proceedings of {L}iverpool {S}ingularities {S}ymposium, {I} (1969/70)},
      PAGES = {286--307},
      SERIES={ Lecture Notes in Math.},
      VOLUME={192},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1971},
      MRCLASS = {57D45 (14E15)},
      MRNUMBER = {0293646},
      MRREVIEWER = {J. M. Boardman},
      ZBLNUMBER = {0221.57016},
      DOI = {10.1007/BFb0066829},
     }
  • [por] I. R. Porteous, "Probing singularities," in Singularities, Part 2, Providence, R.I.: Amer. Math. Soc., 1983, vol. 40, pp. 395-406.
    @incollection {por, MRKEY = {0713263},
      AUTHOR = {Porteous, Ian R.},
      TITLE = {Probing singularities},
      BOOKTITLE = {Singularities, {P}art 2},
      VENUE={{A}rcata, {C}alif., 1981},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {40},
      PAGES = {395--406},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, R.I.},
      YEAR = {1983},
      MRCLASS = {58C27 (57R45)},
      MRNUMBER = {0713263},
      MRREVIEWER = {Felice Ronga},
      ZBLNUMBER = {0523.58010},
      }
  • [prag] P. Pragacz, Thom polynomials and Schur-functions I..
    @misc{prag,
      author={Pragacz, P.},
      TITLE={Thom polynomials and {S}chur-functions {I.}},
      ARXIV={math.AG/0509234},
      YERA={2005},
     }
  • [sigma11] F. Ronga, "Le calcul des classes duales aux singularites de Boardman d’ordre 2," C. R. Acad. Sci. Paris Sér. A-B, vol. 270, p. a582-a584, 1970.
    @article {sigma11, MRKEY = {0263097},
      AUTHOR = {Ronga, Felice},
      TITLE = {Le calcul des classes duales aux singularites de {B}oardman d'ordre 2},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
      VOLUME = {270},
      YEAR = {1970},
      PAGES = {A582--A584},
      MRCLASS = {57.20},
      MRNUMBER = {0263097},
      MRREVIEWER = {J. M. Boardman},
      ZBLNUMBER = {0191.54202},
     }
  • [ronga] F. Ronga, "A new look at Faà de Bruno’s formula for higher derivatives of composite functions and the expression of some intrinsic derivatives," in Singularities, Part 2, Providence, RI: Amer. Math. Soc., 1983, vol. 40, pp. 423-431.
    @incollection {ronga, MRKEY = {0713268},
      AUTHOR = {Ronga, Felice},
      TITLE = {A new look at {F}aà de {B}runo's formula for higher derivatives of composite functions and the expression of some intrinsic derivatives},
      BOOKTITLE = {Singularities, {P}art 2},
      VENUE={{A}rcata, {C}alif., 1981},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {40},
      PAGES = {423--431},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1983},
      MRCLASS = {57R45 (26B12 58C20 58C27)},
      MRNUMBER = {0713268},
      MRREVIEWER = {J. M. Boardman},
      ZBLNUMBER = {0541.58010},
      }
  • [rimanyi] Go to document R. Rimányi, "Thom polynomials, symmetries and incidences of singularities," Invent. Math., vol. 143, iss. 3, pp. 499-521, 2001.
    @article {rimanyi, MRKEY = {1817643},
      AUTHOR = {Rim{á}nyi, Rich{á}rd},
      TITLE = {Thom polynomials, symmetries and incidences of singularities},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {143},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {499--521},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {58K30 (57R45 57R70)},
      MRNUMBER = {1817643},
      MRREVIEWER = {Toru Ohmoto},
      DOI = {10.1007/s002220000113},
      ZBLNUMBER = {0985.32012},
      }
  • [rossmann] W. Rossmann, "Equivariant multiplicities on complex varieties," in Orbites Unipotentes et Représentations, III, , 1989, vol. 173-174, pp. 313-330.
    @incollection {rossmann, MRKEY = {1021516},
      AUTHOR = {Rossmann, W.},
      TITLE = {Equivariant multiplicities on complex varieties},
      BOOKTITLE = {Orbites Unipotentes et Repr{é}sentations, {III}},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {173-174},
      YEAR = {1989},
      PAGES = {313--330},
      ISSN = {0303-1179},
      MRCLASS = {32M05 (32S20 32S70)},
      MRNUMBER = {1021516},
      ZBLNUMBER = {0691.32004},
     }
  • [szenes] Go to document A. Szenes, "Iterated residues and multiple Bernoulli polynomials," Internat. Math. Res. Notices, iss. 18, pp. 937-956, 1998.
    @article {szenes, MRKEY = {1653791},
      AUTHOR = {Szenes, Andr{á}s},
      TITLE = {Iterated residues and multiple {B}ernoulli polynomials},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {1998},
      NUMBER = {18},
      PAGES = {937--956},
      ISSN = {1073-7928},
      MRCLASS = {11B68 (11L99 14D20 52C35)},
      MRNUMBER = {1653791},
      MRREVIEWER = {Hiroaki Terao},
      DOI = {10.1155/S1073792898000567},
      ZBLNUMBER = {0968.11015},
      }
  • [thom] R. Thom, "Les singularités des applications différentiables," Ann. Inst. Fourier, Grenoble, vol. 6, pp. 43-87, 1955–1956.
    @article {thom, MRKEY = {0087149},
      AUTHOR = {Thom, R.},
      TITLE = {Les singularités des applications différentiables},
      JOURNAL = {Ann. Inst. Fourier, Grenoble},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {6},
      YEAR = {1955--1956},
      PAGES = {43--87},
      ISSN = {0373-0956},
      MRCLASS = {53.0X},
      MRNUMBER = {0087149},
      MRREVIEWER = {W. S. Massey},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0075.32104},
      }
  • [voj] Go to document M. Vergne, "Polynômes de Joseph et représentation de Springer," Ann. Sci. École Norm. Sup., vol. 23, iss. 4, pp. 543-562, 1990.
    @article {voj, MRKEY = {1072817},
      AUTHOR = {Vergne, Mich{è}le},
      TITLE = {Polynômes de {J}oseph et représentation de {S}pringer},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {23},
      YEAR = {1990},
      NUMBER = {4},
      PAGES = {543--562},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {17B15 (17B35 22E60)},
      MRNUMBER = {1072817},
      MRREVIEWER = {Anthony Joseph},
      URL = {http://www.numdam.org/item?id=ASENS_1990_4_23_4_543_0},
      ZBLNUMBER = {0718.22009},
      }

Authors

Gergely Bérczi

Mathematical Institute
24-29 St Giles
Oxford OX1 3LB
England

András Szenes

Mathematics Institute
Budapest University of Technology
Budapest 1111
Hungary

Current address:

Section de mathématiques
Université de Genève
2-4 rue du Lièvre
Case postale 64
CH-1211 Genève 4
Switzerland