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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.

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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.

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