### D.15.21 sagbiNormaliz_lib

Library:
  sagbiNormaliz.lib
Purpose:
  Provides an interface for the computation of Sagbi bases. It uses normaliz (version 3.10.5 or higher) for combinatorial computations.

Authors:
  Winfried Bruns, wbruns@uos.de

Overview:
  The library sagbiNormaliz.lib provides functions for the computations of Sagbi bases of subalgebras A of polynomial rings over a field. It is based on normaliz.lib. Its functions compute Sagbi bases with or without the control by Hilbert functions and/or degrees. Hilbert functions and degrees require that bthe ambient polynomial ring is standard graded. (An extension to general positive gradings would not be difficult.)

  In addition to the Sagbi bases it can compute a defining ideal for the algebra A with respect to the given system of generators. (The computation of defining ideals cannot be controlled by Hilbert series.)

  See the papers

  (1) W.Bruns and A. Conca, Sagbi combinatorics of maximal minors and a Sagbi algorithm. J. Symb. Comput. 120 (2024), Article ID 102237, 14 p. (arXiv:2302.14345)

  (2) W. Bruns, A. Conca and F.Lembo, Sagbi bases, defining ideals and algebras of minors.

  Use of this library requires the program Normaliz, version n 3.10.5 or newer, to be installed and the availability of normaliz.lib. You can download both from @uref{https://github.com/Normaliz/Normaliz/releases}. Please make sure that the executable is in the search path or use setNmzExecPath (defined in normaliz.lib).

  The computations of this library require reading Normaliz output files and therefore a file name must be set. The standard file name chosen by the library is NmzSagbiExchange in the current directory. The user can change the name.

Procedures:

> D.15.22.1 sagbiGeneral |  | computes the Sagbi basis of the subalgebra of the current polynomial ring that is generated by the elements of Q. The computation is stopped after at most sagbiMaxRounds rounds if trhe parameter is set. If sorting is set, the computed elements are degrevlex sorted before a round of the algorithm. The optional parameter verb sets the terminal output. Default is 1 = on..
> D.15.22.2 sagbiByDegree |  | computes the Sagbi basis degree by degree until thwe degree bound is reached or the Sagbi basis has been computed compltely.

> D.15.22.3 sagbiHilbControlled

> computes the Sagbi basis up to the degree bound. The Hilbert series of the subalgebra generated by the elements of Q is given by its numerator and denominator as a rational function. HS_denom_algebra lists the exponents g_i in the factors 1 -t^g_i of the denominator. If the degree bound is reached and the optional argument inalCheck is set, the Hilbert series is checked again for completion.

> D.15.22.4 sagbiDefIdealGeneral |  | does the same as sagbiGeneral, but additionally computes as much of a system of generators of the defining idael as it can get before being stopped. Addutionally it returns the list of expressions of the Sagbi elements in terms of the original generators of the aubalgebra, calles the retract.
> D.15.22.5 sagbiDefIdealByDegree |  | does the same as sagbiByDegree, but additionally computes the defining ideal up to the degree set by Sagbi_degree_bound. Addutionally it returns the list of expressions of the Sagbi elements in terms of the original generators of the aubalgebra, calles the retract.

User manual for Singular version 4.4.1, 2025, generated by texi2html.