SINGULAR-1.0: a computer algebra system for singularity theory, algebraic geometry and commutative algebra

CENTER PROBLEM FOR THIRD-ORDER ODEs

“…Because the focus quantities tend to be very large and contain several hundreds (or even thousands) of monomials, the decomposition precedure requires more specialized computer algebra package such as Singular [Greuel et al (2005)], CALI [Gräbe(1995)] or Macaulay [Grayson & Stillman()]. We performed our computation with minAssGTZ of Singular (library primedec.lib [Decker et al (2001)]) in order to find the minimal associated primes of a polynomial ideal by means of the Gianni-Trager-Zacharias method [Gianni et al (1988)].…”

Computing the Distance Distribution of Systematic Nonlinear Codes

“…With this notation, our previous estimates for C can be written as C ≃ 2 3k . We tested our problem using different computer algebra systems (Magma 2.10.13 [MAG], Polybori 0.3.1 [pol], Singular 2.0.6, Singular 3.0.4 [GPS07]). For any system the time needed has a behavior of kind 2 αk , with α depending on the system.…”

The Ernst equation and ergosurfaces

“…In the case b ≡ 0 this issue is easy to handle, by noting that a can then always be made to vanish by a redefinition of t. Now, the length g(∂ ϕ , ∂ ϕ ) of the Killing vector ∂ ϕ , generating rotations around the axis, is a smooth-hence locally bounded-function on the spacetime. But g ϕϕ = ρ 2 f −1 by (2.1) so, in the static case, zeros of f with ρ 0 = 0 cannot correspond to ergosurfaces in spacetime 8 .…”

“…It is straightforward, using the Cauchy-Riemann equations, to check that functions of this form satisfy (5.13)-(5.14), for all k ∈ N. (In fact, both the leftand right-hand sides then vanish identically.) Those solutions have been found by inspection of the solutions found by MAPLE for k = 2 and by SINGULAR [7,8] for k = 3 and 4. In fact, both the SINGULAR-generated solutions, as well as our remaining computer experiments using SINGULAR, played a decisive role in our solution of the problem at hand.…”