Singular

Abstract

What is SINGULAR? SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory. SINGULAR's main computational objects are polynomials, ideals and modules over a large variety of rings, including important non-commutative rings. SINGULAR features one of the fastest and most general implementations of various algorithms for computing standard resp. Gröbner bases. Furthermore, it provides multivariate polynomial factorization, resultant, characteristic set and gcd computations, syzygy and free-resolution computations, numerical root-finding, visualization, and many more related functionalities. What is new? Gröbner bases over ringsThe Gröbner basis routines were modified to work for polynomial rings with coefficients from a ring like Z or Z/m instead of coming from a field. Coefficient rings of the form Z/(2 n ) are important for the applications, for example in proving the arithmetic correctness of data paths in System-on-Chip modules:We start with a set of equations G j , j = 1, . . . , m given by polynomials f j ∈ Z[X], X a set of variables, which are of the form G j :For the variables rk ∈ X in this equation we assume r ( j) i = a (l) k for 1 ≤ l ≤ j and all i, k. We call the variables a ( j) i inputs and r ( j) i outputs of G j . For every proof goal, we obtain an additional polynomial g depending on a subset of variables {a 1 , . . . , a t } ⊂ X and need to check whether g(a 1 , . . . , a t ) = 0 mod 2 n for all solutions of the set of equations {G j }. Example 1 A k-bit comparator of operands a and b is modeled by the polynomialDenote the set of all solutions to {G j } as V ({G j }). Analogously let V (g) be the set of all roots of g. We create an equivalent variety subset problem V ({h i }) ⊂ V (g) where h i and g are polynomials over a single ring Z/2 N with appropriate N. This problem can be solved effectively using Gröbner basis techniques, such as normal form computations.This methods was used to verify a multiply-accumulate-unit (which shall compute a + b * cmod2 6 4) and was much faster than other methods.Additional to the definition of Gröbner basis over rings (a generating set of polynomials for the ideal, whose leading terms generate the leading term ideal) we need the notion of a strong Gröbner basis: a Gröbner basis, where the leading term of each element of the ideal is divisible by an element of the strong Gröbner basis.SINGULAR computes a strong Gröbner basis with a variant of the algorithm of Buchberger.180

On splice quotients of the form {zn = f (x,y )}

“…Then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widetilde{V}(x,y)=-\frac{1}{c}(x^3+by^2)$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widetilde{M}_2(x,y)=y$\end{document}. Equation (3.5) in this case is For b = c = 1, the computer algebra system Singular (2, 3) confirms that the plane curve singularity defined by y 5 + ( x 3 + y 2 ) 2 = 0 is irreducible, and has two topological pairs, (2, 3) and (2, 15), as we expect. Note that changing n would not make any difference in the computation, as long as n is divisible by 5 and relatively prime to 2 and 3.…”

Algebraic and Combinatorial Algorithms for Translinear Network Synthesis

“…The computer algebra system Singular [GPS01] has been used to compute I N and to perform the elimination for each formal network in the catalog with 7 or less transistors.…”

“…Another direction of concept extension is the extension for MOS translinear circuits, see The algorithms presented in this thesis for building and searching a catalog of translinear network topologies have been implemented using C++, Singular [GPS01], and Mathematica [Wol99].…”

Can string theory predict the Weinberg angle?

“…To this end, we used the computer algebra system SINGULAR [22]. We present the details of the calculation in Appendix A.…”

“…(vi) If there are no solutions to the constraint equations, this rigorously proves the absence of a U(1) that may play the role of hypercharge. In the cases where solutions exist, we determine them numerically using Laguerre's algorithm as implemented by SINGULAR [22]. Equation (6) then gives the corresponding hypercharge direction, for which we can explicitly check whether our criteria are satisfied or not.…”