In Maple, to take the exponent of a term of the form $x^a$, we can define 

    Exponent := proc(x::`^`) 

    local r; 

    r := op(2, x); 

    return(r);

    end proc:

This also works for a term of the form $Y[1,3]^{-1/3}$.

But this does not work for taking exponent of a term of the form $Y[1,3]$. How to define a function $f$ in Maple, such that $f(Y[1,3]^{a}) = a$ for any $a \in \mathbb{Q}$? Thank you very much.

I am trying to use type(sqrt(t)*Y[1, -3]*Y[2, -6], monomial) to check if an expression is a monomial. But it returns false which is not correct. How to check that an expression in Maple is a monomial? Thank you very much.

I have a question about Hilbert Series of the quotient of an algebra by a non-homogenous ideal. 

In maple help: https://www.maplesoft.com/support/help/maple/view.aspx?path=Groebner%2FHilbertSeries, it is said that 

The algorithms for HilbertSeries and HilbertPolynomial use the leading monomials of a total degree Groebner basis for J. Here J is an ideal.

I checked the following example. Let 

J:= [(x[2]-x[1])*(x[2]-x[3]), (x[4]-x[1])*(-x[5]+x[4]), (x[4]-x[2])*(x[4]-x[6]), (-x[3]+x[5])*(x[5]-x[6]), x[1]^3-1, x[2]^3-1, x[3]^3-1, x[4]^3-1, x[5]^3-1, x[6]^3-1];

and

t1:=map(LeadingMonomial, l, grlex(x[1], x[2], x[3], x[4], x[5], x[6], x[7], x[8], x[9], x[10], x[11], x[12]));

Then t1 is the ideal generated by the following monomials.
t1 := [x[1]*x[2], x[1]*x[4], x[2]*x[4], x[3]*x[5], x[1]^3, x[2]^3, x[3]^3, x[4]^3, x[5]^3, x[6]^3]

Using the commands:

l2:={x[1],x[2],x[3],x[4],x[5],x[6],x[7],x[8],x[9],x[10],x[11],x[12]};

factor(HilbertSeries(J, l2, t));

factor(HilbertSeries(t1, l2, t));

We obtain Hilbert(J,t)=(t^2+t+1)*(5*t^4+11*t^3+11*t^2+5*t+1)
Hilbert(t1, t) = (t^2+t+1)*(3*t^2+3*t+1)*(2*t^2+2*t+1)

These two polynomials are not equal. How does maple compute Hilbert Series for the quotient of an algebra by a non-homogenous ideal? Thank you very much.

I want to define a function f in Maple such that $f(x[1,2]^{-1/3})=\{1,2\}, f(x[1,2])=\{1,2\}, f(x[1])=\{1\}, f(x[1]^{1/3}) =\{1\}$, f(x)=empty set. How could I do this in Maple? Thank you very much.