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export { "cyclic" }
cyclic = method()
cyclic (ZZ,Ring) := (n,kk) -> (
R := kk[vars(0..n-1)];
toList apply(1..n-1, d-> sum(0..n-1,
i -> product(d, k -> R_((i+k)%n))
)) | {product gens R - 1}
)
beginDocumentation()
doc ///
Key
cyclic
(cyclic,ZZ,Ring)
Headline
an example of a square polynomial system
Usage
cyclic(n,kk)
Inputs
n:ZZ
the number of variables
kk:Ring
the coefficient ring
Outputs
:List
of polynomials in the system
Description
Text
{\bf Cyclic n-roots} is a popular benchmark problem.
It has finitely many solutions iff n is square free.
In this case the number of solutions is less than the Bezout bound.
This system was solved in May 2020, using @TO solveSystem@ in Macaulay2 v1.15
with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 70 solutions found in 2.737 seconds with 5 variables.
Example
cyclic(5,QQ)
///
-- test disabled, because it will fail on slightly slower machines
-- TEST ///
-- F = cyclic(5,CC_53)
-- assert(
-- first timing (sols = solveSystem F;)
-- < 3
-- )
-- assert(#sols==70)
-- ///
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