I am Seonhwa Kim, a mathematical researcher in Korea. Recently, I have extensively used Maple to compute character varieties of 3-manifolds. Several months ago, I obtained some strange results in Maple which implies a contradiction in theory. I have been struggling with these issues since it is usually about enormous polynomial systems. Eventually, I could figure out that the issues are caused by a defect in Maple and were able to construct a minimal working example to produce wrong computations in Maple. I am writing this post to report them.
This is mainly about the PolynomialIdeal package. Along with the documentation in Maple, If an ideal J is radical, PrimeDecomposition and PrimaryDecomposition should have the same result. But, as we see the following, the result of PrimeDecomposition and PrimaryDecomposition are different although J is a radical ideal.
The problem seems to be that the PrimaryDecomposition command in Maple sometimes produces incorrect results.
We can compute the primary decomposition of J by hand. It should be <x> and <y, x-1>.
I double-checked this by the other software;Macauley2, Singular, and Magma, for example, you can see it as follows.
Secondly, not only for PrimaryDecomposition but also PrimeDecomposition may produce an incorrect result.
Here is a minimal working example.
Maple tells us a compatible result of prime and primary decomposition of a radical ideal J.
But the first component of J, < b-1, c-a+1 >, contains the third component < a, b-1, c+1 >.
It contradicts with the definition of Primary decomposition. So the correct answer should be < b - 1, c - a +1 >, <a,b,c>.
I also checked that Macaluey2, Singular and Magma. They all say that my hand computation is correct. as follows.
I have used Maple 2017 by the license of my institute (Korea Institute for Advanced Study).
When I noticed these defects, I thought it would be fixed in the newest Maple version.
So, I have tried my examples by Maple2019 free trial, but It also has the same problem.
I guess this problem is not reported or recognized yet.
I hope this problem will be fixed as soon as possible.
Thank you for attention.