@Markiyan Hirnyk 

Have you tried the example?
Please note that exactly the same text appears in the help page of LinearAlgebra[SmithForm].

P.S. You are free to use it for complex coefficients. I will not.

BTW, MatrixPolynomialAlgebra:-SmithForm  calls  LinearAlgebra:-SmithForm
so everything should be clear.

@Markiyan Hirnyk 

It does not work for complex coefficients. Try
A := Matrix(3, 3, [[(27*I)*x+27, 99*x-99*I, 92*x^2-92-(184*I)*x], [(8*I)*x+8, 29*x-29*I, -31*x^2+31+(62*I)*x], [(69*I)*x+69, 44*x-44*I, 67*x^2-67-(134*I)*x]]);

The result should be:

@acer 

Yes, the solution is shorter and cleaner.
My excuse for the longer one is that I prefer to use evala for algebraic numbers/expressions (RootOfs of polynomials) whenever possible.

@brian bovril 

Assume N is odd.
Let A[k] be the vertices on the unit circle of a regular polygon, k=0..N-1.
The line L[k] is tangent to the unit circle at A[k].
The furthest point of intersection is at the distance 1/sin(Pi/2/N) from O
(e.g. intersecting L[0] with L[(N-1)/2]).
The coefficient 1.25 was used to have all the intersections inside the circle with radius R.
a[k], b[k] are the intersections of L[k] with the circle of radius R.
d appears easily if we draw a picture and use some trigonometry.

I obtained the result using solve with series.
Actually also some manipulations by hand because series (and asympt) has some bugs. So, other terms are possible in the expansion:
s =

Edit.

It can be shown that asymptotically:
s = 1/2 * (3*Pi)^(2/3) * r^(1/3)   ≈  2.23 * r^(1/3)

@Preben Alsholm 

It also works for Digits=50.
But for Digits=30 it starts increasing and decreasing Digits and never ends.

Edit. The value for Digits should not be a problem because theoretically Digits will be increased by fsolve if needed.

I never said that Maple has problems only for essential singularities.

To compute a limit could be a very complex task; sometimes almost an "art".
When the expression involves an essential singularity, the result given by Maple has big chances to be wrong.

Unfortunately the help file does not say a word about it, but I am sure that the designers (or at least some of them) are aware of this problem. In these cases, there should appear many "FAIL" answers but this would not be a strong point in marketing.

Here are some real examples (i.e. without a complex direction):

limit((1/2+sin(x))^(1/x), x=infinity);
            1

limit((1+sin(x))^(1/x), x=infinity);
           1

limit( (1+x*cos(x))/(2+x*cos(x)), x=infinity);
           1

@Markiyan Hirnyk 
It seems that you don't understand.
The existence of this f shows that for a CAS it will be very hard to decide whether e.g. the following simple function
g: = sin(exp(-1/z^2+z))*exp(z)-exp(-z)/z^2;
has an essential singularity at z=0 or not.

@Axel Vogt 

The construction of such an f is very technical. It can be done using a deep theorem in complex analysis:
Rosay J.-P., Rudin W. - Arakelian's Approximation Theorem. Amer. Math. Monthly, 96 (1989), 432-434
(the construction itself is not in the article).

@nm 
For a code snippet click the < > icon (near the A icon).

A computation method used by Maple looks very concrete to me.
But the question was not directed exclusively to you.

In this thread I had a concrete question, not an answer. Wasn't this clear enough?

This is just like saying that Riemann Hypothesis is a piece of cake because everyone knows the definition for the zero of a function.