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On the Wronskian combinants of binary forms. (English) Zbl 1115.13010

This is a technical paper in classical invariant theory. Given a sequence \(\mathcal A\) of binary forms, the authors construct the set of Wronskian combinants of \(\mathcal A\). They show that the span of \(\mathcal A\) can be recovered from the combinants as the solution of an SL(2)-invariant differential equation. Their main result characterizes those sequences of binary forms that can arise as Wronskian combinants, and they deduce several identities relating Wronskians to transvectants.

MSC:

13A50 Actions of groups on commutative rings; invariant theory
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[1] A. Abdesselam, J. Chipalkatti, The bipartite Brill-Gordan locus and angular momentum, Transform. Groups, (math.AG/0502542; A. Abdesselam, J. Chipalkatti, The bipartite Brill-Gordan locus and angular momentum, Transform. Groups, (math.AG/0502542 · Zbl 1103.14030
[2] Abhyankar, S. S., Invariant theory and enumerative combinatorics of Young tableaux, (Mundy, J.; Zisserman, A., Geometric Invariance in Computer Vision. Geometric Invariance in Computer Vision, Artificial Intelligence (1992), MIT Press: MIT Press Cambridge, MA), 45-76
[3] J. Chipalkatti, On the invariant theory of the Bezoutiant, Beiträge Algebra Geom., (math.AG/0406410; J. Chipalkatti, On the invariant theory of the Bezoutiant, Beiträge Algebra Geom., (math.AG/0406410 · Zbl 1112.13009
[4] Dolgachev, I., (Lectures on Invariant Theory. Lectures on Invariant Theory, London Mathematical Society Lecture Notes, vol. 296 (2003), Cambridge University Press) · Zbl 1023.13006
[5] Fulton, W.; Harris, J., (Representation Theory, A First Course. Representation Theory, A First Course, Graduate Texts in Mathematics (1991), Springer-Verlag: Springer-Verlag New York) · Zbl 0744.22001
[6] Glenn, O., The Theory of Invariants (1915), Ginn and Co.: Ginn and Co. Boston, (PG) · JFM 45.0240.01
[7] Gordan, P., Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist, J. Reine Angew. Math., 69, 323-354 (1868), (GDZ) · JFM 01.0046.01
[8] Gordan, P., Ueber Combinanten, Math. Ann., 5, 95-122 (1872), (GDZ) · JFM 04.0061.01
[9] Grace, J. H.; Young, A., The Algebra of Invariants, 1903 (1962), Chelsea Publishing Co.: Chelsea Publishing Co. New York, (reprinted) (MiH)
[10] Hartshorne, R., (Algebraic Geometry. Algebraic Geometry, Graduate Texts in Mathematics (1977), Springer-Verlag: Springer-Verlag New York) · Zbl 0367.14001
[11] Hazlett, O., A symbolic theory of formal modular covariants, Trans. Amer. Math. Soc., 24, 4, 286-311 (1922), (available via JStor) · JFM 50.0058.01
[12] Iarrobino, A., Deforming complete intersection Artin algebras. Singularities, Part 1, Proc. Sympos. Pure Math., 40, 593-608 (1983)
[13] Iarrobino, A.; Kleiman, S., (Appendix C in Power Sums, Gorenstein Algebras and Determinantal Loci (by A. Iarrobino and V. Kanev). Appendix C in Power Sums, Gorenstein Algebras and Determinantal Loci (by A. Iarrobino and V. Kanev), Springer Lecture Notes in Mathematics, vol. 1721 (1999), Springer-Verlag: Springer-Verlag Berlin)
[14] Kleiman, S., Les théorèmes de finitude pour le foncteur de Picard, (SGA 6, Exposé XIII. SGA 6, Exposé XIII, Springer Lecture Notes in Mathematics, vol. 225 (1970)) · Zbl 0227.14007
[15] Kung, J. P.S.; Rota, G.-C., The invariant theory of binary forms, Bull. Amer. Math. Soc., 10, 1, 27-85 (1984) · Zbl 0577.15020
[16] M. Meulien, Sur les invariants des pinceaux de quintiques binaires. Thèse de Doctorat, Université de Versailles Saint-Quentin-en-Yvelines, 2002; M. Meulien, Sur les invariants des pinceaux de quintiques binaires. Thèse de Doctorat, Université de Versailles Saint-Quentin-en-Yvelines, 2002
[17] Meulien, M., Sur les invariants des pinceaux de formes quintiques binaires, Ann. Inst. Fourier (Grenoble), 54, 1, 21-51 (2004) · Zbl 1062.14060
[18] Meulien, M., Sur la complication des algèbres d’invariants combinants, J. Algebra, 284, 1, 284-295 (2005) · Zbl 1067.14042
[19] Olver, P., (Classical Invariant Theory. Classical Invariant Theory, London Mathematical Society Student Texts (1999), Cambridge University Press)
[20] Pasch, M., Note über die Determinanten, welche aus Functionen und deren Differentialen gebildet werden, J. Reine Angew. Math., 80, 177-182 (1875), (GDZ) · JFM 07.0079.02
[21] Salmon, G., Lessons Introductory to the Modern Higher Algebra (1964), Chelsea Publishing Co.: Chelsea Publishing Co. New York, (reprinted)
[22] Schmidt, F. K., Die Wronskische Determinante in beliebigen differenzierbaren Funktionenkörpern, Math. Z., 45, 62-74 (1938), (GDZ) · Zbl 0020.10201
[23] Sturmfels, B., (Algorithms in Invariant Theory. Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation (1993), Springer-Verlag: Springer-Verlag Wien, New York)
[24] Sylvester, J. J., On the calculus of forms, otherwise the theory of invariants, (Collected Mathematical Papers, vol. 1, No. 54 (1904), Cambridge University Press). (Collected Mathematical Papers, vol. 1, No. 54 (1904), Cambridge University Press), Cambridge and Dublin Math. J., 8, 411-422 (1853), Reprinted from:
[25] Verdier, J.-L., (Applications harmoniques de \(S^2\) dans \(S^4\), II. Harmonic mappings, twistors, and \(\sigma \)-models (Luminy, 1986). Applications harmoniques de \(S^2\) dans \(S^4\), II. Harmonic mappings, twistors, and \(\sigma \)-models (Luminy, 1986), Adv. Ser. Math. Phys., vol. 4 (1988), World Sci. Publishing: World Sci. Publishing Singapore), 124-147
[26] Weyman, J., Cohomology of Vector Bundles and Syzygies (2003), Cambridge University Press · Zbl 1075.13007
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