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edit approved Sep 3, 2014 at 18:18
user26857
edit approved Sep 3, 2014 at 18:18
user26857
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Do you assume the ideal is homogeneous? No power of S$S$ belong to (X,S-1)$(X,S-1)$.

If you assume that the ideal is homogeneous, then S^m$S^m$ is in the ideal (f,g)$(f,g)$ for m=deg f+deg g-1$m=\deg f+\deg g-1$. Indeed all homogeneous polynomials of that degree are in the ideal (f,g)$(f,g)$. This is a simple Hilbert series computation, just use the fact that the Koszul complex resolves K[X,S]/(f,g)$K[X,S]/(f,g)$ and use it to compute the Hilbert series of K[X,S]/(f,g)$K[X,S]/(f,g)$.

Do you assume the ideal is homogeneous? No power of S belong to (X,S-1).

If you assume that the ideal is homogeneous, then S^m is in the ideal (f,g) for m=deg f+deg g-1. Indeed all homogeneous polynomials of that degree are in the ideal (f,g). This a simple Hilbert series computation, just use the fact that the Koszul complex resolves K[X,S]/(f,g) and use it to compute the Hilbert series of K[X,S]/(f,g).

Do you assume the ideal is homogeneous? No power of $S$ belong to $(X,S-1)$.

If you assume that the ideal is homogeneous, then $S^m$ is in the ideal $(f,g)$ for $m=\deg f+\deg g-1$. Indeed all homogeneous polynomials of that degree are in the ideal $(f,g)$. This is a simple Hilbert series computation, just use the fact that the Koszul complex resolves $K[X,S]/(f,g)$ and use it to compute the Hilbert series of $K[X,S]/(f,g)$.

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user48585
user48585
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Do you assume the ideal is homogeneous? No power of S belong to (X,S-1).

If you assume that the ideal is homogeneous, then S^m is in the ideal (f,g) for m=deg f+deg g-1. Indeed all homogeneous polynomials of that degree are in the ideal (f,g). This a simple Hilbert series computation, just use the fact that the Koszul complex resolves K[X,S]/(f,g) and use it to compute the Hilbert series of K[X,S]/(f,g).