Suppose we look at the $\mathbb {C}$ module $\mathbb{C}[x_1,\dots,x_n]=\oplus{S_i}$ where $S_i$ are the polynomials of degree $i$. Then we look at a subring $R$ (also a $\mathbb {C} $ module) generated by polynomials $p_1,\dots,p_m$ which have some algebraic relations among them. We grade $R=\oplus{R_i}$ where $R_i=R\cap S_i$. Suppose we know an upper bound on the degree that the polynomials $p_i$ can have. Is there an obvious way to transform this into a bound on the Hilbert regularity of $R$? My guess would be that they are the same. Any help would be appreciated.

## 1 Answer

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Let $I=(x^{10},y^{10})$. The maximum degree of a generator is 10. The Hilbert polynomial is 0, but the function does not attain this value until degree 19, so your guess isn't correct. It *is* related to the degree of the generators of the Gröbner basis; see for instance *Using Algebraic Geometry* (Cox, Little, O'Shea).

An illustration with Sage:

```
sage: R.<x,y> = QQ[]
sage: I = R.ideal([x^10,y^10])
sage: I.hilbert_polynomial()
0
sage: I.hilbert_series()
t^18 + 2*t^17 + 3*t^16 + 4*t^15 + 5*t^14 + 6*t^13 + 7*t^12 + 8*t^11 + 9*t^10 + 1
0*t^9 + 9*t^8 + 8*t^7 + 7*t^6 + 6*t^5 + 5*t^4 + 4*t^3 + 3*t^2 + 2*t + 1
```

Notice the series' coefficients don't become 0 until degree 19.