Consider the ideal $I=\langle x_0^2+x_1^2+x_2^2+x_3^2,x_0x_1+x_2x_3\rangle \subseteq k[x_0,x_1,x_2,x_3]$ for $k=\mathbb{F}_{32749}$. I am trying to calculate the dimension of the projective variety generated by this ideal.
Following the book Ideals, Varieties, and Algorithms, I first note that the generators $x_0^2+x_1^2+x_2^2+x_3^2$ and $x_0x_1+x_2x_3$ are a Gröbner basis for $I$. Thus, $\langle LT(I)\rangle =\langle x_0^2,x_0x_1\rangle$ using Lex order with $x_0>x_1>x_2>x_3$.
Then I count the number of monomials of degree $s$ not in $\langle LT(I) \rangle$ to get the Hilbert function.
The number of monomials of degree $s$ in $4$ variables is $\binom{s+3}{s}$.
The monomials of degree $s$ in $\langle LT(I) \rangle$ have the form, $$x_0^2f$$ or $$x_0x_1g$$ where $f,g$ have degree $s-2$. The number of monomials of degree $s-2$ is $\binom{s+1}{s-2}$.
There is double counting if we subtract this number twice, so we have to count the overlap, that is when, $$x_0^2f=x_0x_1g,$$ in other words, $$f=x_1\alpha$$ and $$g=x_0\alpha$$ for some monomial of degree $s-3$, $\alpha$. The number of monomials of degree $s-3$ is $\binom{s}{s-3}$. Hence, the Hilbert function is, $$HF=\binom{s+3}{s}-2\binom{s+1}{s-2}+\binom{s}{s-3}=\frac{s^2+5s+2}{2}$$ which is the Hilbert Polynomial for large enough $s$.
When using Macaulay2, I get the the Hilbert polynomial is, $$HP(s)=4s.$$
Where am I going wrong?