When I use Maple to solve some complex polynomial system and trying different methods, for instance, using engine = traditional, it will sometimes lose or fail to find the solutions. In the document, it says Maple will convert to engine = groebner when it does not find solutions using traditional engine. As far as I know it is using resultant to eliminate variables, and the resultant is computed by pseudo remainder. The leading coefficient of the divisor is multiplied during each division to ensure that we include the case that it is zero.

But why this procedure will lose solutions sometimes? What makes Maple abandon this method?

I intend to use LinearAlgebra package to do some calculations. I want to compute the basis for large Matrices. My discovery is that the linalg[kernel] command, which the document claims is deprecated, could do such computation significantly faster than the LinearAlgebra[NullSpace] command. For a 200 x 500 large random matrix, linalg[kernel] clocked 33 secs, while the LinearAlgebra[NullSpace] takes 200 secs, as shown in the worksheet NullSpace_vs_kernel.mw.

I wanna know what makes the difference, or is there a misuse for LinearAlgebra[NullSpace].

 Based on the type of coefficients in the linear equations, SolveTools[Linear] provide several method including method = Rational, Polynomial etc. 
The Polynomial method of SolveTools[Linear] however, cannot be directly called by SolveTools[Linear]. The interpreter complains that no such methd called "Polynomial".

This could only be solved if I import the SolveTools package in advance and call Linear.

I hope Maple could solve this issue. 

So just like the title illustrates, I found a paper authored by Gary Nicklason in 2022: Autonomous Planar Systems of Riccati Type and in the last section it mentioned about a class of Abel ODE, which belongs to AIA(Abel Inverse Abel) class. It is of First kind and the inverse of it(by swapping variables) is of second kind.

While the first kind is solvable in terms of Airy function, the inverse of it along with its equivalence class is not solvable by the existing dsolve.

I have tested it in my worksheet Nicklason_equation.mw. So is it possible to add this class into the dictionary for solvable Abel ODE, or, maybe there are some bugs within the internal procedure of dsolve, which results in failure for catching the solvable candidates?

I was trying to solve a system of eight cubic equations, with eight variables. Note that the particular solutions should exist and my goal is to find all  possible solutions using solve. However, when executing, the solve keeps running for a whole day and did not throw any results. I also set the parameter infolevel[solve] to be 3 and find out that it is stuck in the step "GroebnerBasis: computing a factored plex basis using Groebner[Solve]". Can anyone tell me how to deal with that? Here's the Maple file.

solve_test.mw