Wishlist from talks of Levandovskyy, Andres and Studzinski

[levandov]

1. Singular:Plural, Wishlist and future plans

Hilbert polynomial of graded ideals and modules

Hilbert-driven Gr\"obner basis

modular algorithms for Gr\"obner basis, Gr\"obner trinity, Gr\"obner basics and so on

a command like trinity: given M, returns std(M),syz(M),lift(M) at once

implement Gel'fand--Kirillov dimension in kernel (now in a library gkdim.lib)
implement division for Plural; discuss usage of lift in its internals

special Gr\"obner basis for homogenized objects: compute $(I^h):h{\infty}$ by extracting $h$-content of any new element, entering the basis. Note: this can be used alone, that is not in conjunction with Hilbert-driven Groebner basis

bisyzygies and biresolution (algorithms known)

Gr\"obner-free assistance in elimination (linear algebra, see the talk of Daniel Andres)

standard basis and division algorithm for algebras like $\K(q_{ij})[x]_{\langle x \rangle} \langle \partial \mid \partial_j x_i = q_{ij} x_i \partial_j + \delta_{ij} \rangle$ (local ($q$-)Weyl algebras)

2. Locapal: Future plans and wishes

Gr\"obner trinity: syz (under testing), lift are needed

user-friendly interface concept \& natural presentation of objects
need: antiblock ordering (ring def/ringlist like $\omega(<_1,\prec_2)$)
need: fast and furious linear algebra over $\K(X)$
crucial need: ehnanced $\gcd$ over $\K[X]$ (e.g. content)
Algorithms to be implemented

free resolutions, \texttt{modulo}, left-right transfer (for hom. algebra)
closure properties (annihilators of a sum/product/... of functions)
integration (Zeilberger, Takayama, Chyzak and other algorithms with plenty of applications), which is used for summation as well
factorization of operators in $\K(X)\langle Y \rangle$

3. Letterplace. Future plans and wishes

Definition: Finitely presented algebra (f.p.a) $:=$ $\K\langle X \rangle/T$, \ $T$ is a twosided ideal.

One-sided Gr\"obner bases over finitely presented algebras
Gr\"obner basics for one- and two-sided modules over f.p.a
One- and two-sided syzygies and resolutions over f.p.a.
Hilbert function and dimension (like Gel'fand-Kirillov)
Homological algebra (need resolution, modulo, opposite structure)

Question to SAGE: interface for manipulating expressions containing words in a finite alphabet with coefficients, Letterplace as back-engine.

Distant future: vector enumeration (code by Steve Linton) under SAGE umbrella?

Since letterplace computations take place in a commutative ring indeed, when will it be possible to incorporate coefficients over a ring like $\Z[a,b,c]$? There are applications needing this (e.g. from Malle, Mueller)

[levandov]

I cannot upload the slides of D. Andres and G. Studzinski due to the size limit on the upload (the files are 540 and 340 KB big).

[levandov]

The list is clearly to big and hence we proceed by decomposing this list into single requests.