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A new basis for the U-invariants of binary forms

Jafari, A ; Sharif University of Technology | 2023

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  1. Type of Document: Article
  2. DOI: 10.1080/00927872.2022.2096893
  3. Publisher: Taylor and Francis Ltd , 2023
  4. Abstract:
  5. Let K be a field of characteristic zero. The K-vector space Vn of all binary forms (Formula presented.) of degree n with coefficients in K carries a natural action of the group (Formula presented.) via substitutions of variables x and y. The invariant polynomials (Formula presented.) under the action of certain subgroups of (Formula presented.) were studied intensively in the 19th century. If U is the subgroup of the upper triangular unipotent matrices, with the aid of heavy computer calculations, we know the algebra of U-invariant polynomials in (Formula presented.) only for (Formula presented.) It appears to be a hopeless task to get much further along these lines. However, it is known that after inverting the U-invariant a 0, the algebra has a very special form, namely it is generated by algebraically independent polynomials (Formula presented.) together with (Formula presented.) In this note, we give an explicit set of such polynomials (Formula presented.) which are of degrees 2 and 3. We also extend these results to U-invariants of several binary forms. © 2022 Taylor & Francis Group, LLC
  6. Keywords:
  7. Binary forms ; Invariant theory ; U-invariants ; K-vector space ; Invariant polynomials
  8. Source: Communications in Algebra ; Volume 51, Issue 1 , 2023 , Pages 248-253 ; 00927872 (ISSN)
  9. URL: https://www.tandfonline.com/doi/abs/10.1080/00927872.2022.2096893