4
$\begingroup$

Let $A=\mathbb C[x_0,\dots,x_{m-1}]$ be the polynomial ring on $m$ variables.

Define $X(u)=\sum_{i=0}^{m-1} x_i u^{i+1}$ and denote by $(X(u)^r)_n$ the coefficient of $u^n$ in the expansion of the $r$-th power of $X(u)$, i.e $X(u)^r$.

Set $I=\langle (X(u)^{r})_s\mid 1\le r \le m+1, \ s\ge m+1 \rangle$.

I am trying to find a basis for $A/I$ and I am guided by some questions:

1) Is that possible to calculate the dimension of $A/I$? In fact, I am happy if I find a way to prove that it is at most $2^m$.

2) What is a Gröbner basis for $I$?

3) What is a linear basis for $A/I$?

Any help are welcome!


Added: If we let $m=2$, then $X(u) = x_0u +x_1u^2$ $X(u)^2 = x_0^2u^2 + 2x_0x_1u^3 + x_1^2 u^4$ $X(u)^3 = x_0^3 u^3+ 3x_0^2x_1u^4 + 3 x_0x_1^2u^5+ x_1^3u^6$ so then $I$ would be generated by $\{x_0x_1,\ x_1^2, \ x_0^3,\ x_0^2x_1, \ x_0x_1^2,\ x_1^3 \}.$ Therefore we conclude that $I$ is the ideal generated by $\{ x_0x_1,\ x_1^2, \ x_0^3 \}$, so that $A/I$ is a $\mathbb C$-space with basis given by the image (with respect to the natural projection) of $\{ 1, \ x_0, \ x_0^2, \ x_1 \}$.

  • 0
    In commutative algebra *dimension* of a ring stands for the Krull dimension. Since you want in (1) the dimension of $A/I$ as a $\mathbb C$-vectorspace, maybe it would be good to say it explicitely.2012-12-28

1 Answers 1

1

So, if you happen to have access to MAGMA, you can run this code.

m := 5; P<[x]> := PolynomialRing(RationalField(), m+1);  p := P!0; for i in [1..m] do     p := p + x[i]*x[m+1]^i;     end for;  Igen := []; for r in [1..m+1] do     C := Coefficients(p^r,x[m+1]);     for s in [m+2..#C] do         Append(~Igen,C[s]);         end for;     end for;  I := ideal;  GroebnerBasis(EasyIdeal(I));  Q := P/I;  Dimension(Q); 

Using this I've found that $\text{dim}(A/I)=2^m$ exactly for $m\leq 9$ and computed the corresponding Groebner bases and quotient rings. I am not sure how this could be proven for all $m$. Aside from actually computing it, I don't know if there is any way to get the basis of $I$ or $A/I$ in general, as the output seems quite complicated. MAGMA uses the $\rm F_4$ algorithm to find Groebner bases, which seems to be the fastest method available at this point in time. If you try to implement a Groebner basis algorithm yourself, however, I would suggest Mutant XL since it is intuitively very easy and still pretty fast.

  • 0
    Thanks for the suggestion by using MAGMA.$I$wrote a procedure to calculate the Grobner basis for $I$ and the linear basis for $A/I$ and I am obtaining the answer until $m=9$ too. I will leave open the question to see if anyone can suggest anything else.2012-12-27