Count the number of homogeneous polynomials - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:46:17Z http://mathoverflow.net/feeds/question/57870 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57870/count-the-number-of-homogeneous-polynomials Count the number of homogeneous polynomials Moduli 2011-03-08T18:33:30Z 2011-03-09T07:19:19Z <p>Is there a general way of counting the number of homogeneous polynomials of certain degree in a complex projective space or a weighted complex projective space, mod the ideal generated by some homogeneous polynomials with smaller degrees? </p> <p>As an example, consider the degree 18 homogeneous polynomials in $W\mathbb{P}_{[2,2,2,4]}^3$, mod the ideal generated by two degree 8 homogeneous polynomials $P_1=x^4_1$ and $P_2=x_2^4$, where $x_1$ and $x_2$ are the first and second coordinates of $W\mathbb{P}_{[2,2,2,4]}^3$. I can count the number of equivalent classes by directly listing all of such homogeneous polynomials; I would like to know if there is a more general and efficient way of doing this.</p> <p><em>Edit</em>: to avoid possible confusion, I have replaced "polynomial" by "homogeneous polynomial".</p> http://mathoverflow.net/questions/57870/count-the-number-of-homogeneous-polynomials/57874#57874 Answer by J.C. Ottem for Count the number of homogeneous polynomials J.C. Ottem 2011-03-08T19:01:38Z 2011-03-09T07:19:19Z <p>I guess what you are after is the Hilbert function of ideal. More precisely, if you have a multigraded polynomial ring $k[x_1,\ldots,x_n]$ and the ideal is given by $I=(f_1,\ldots,f_r)$, the Hilbert function is simply the number $$h(\alpha)=\dim_k (k[x_1,\ldots,x_n]/I)_\alpha.$$Here $\alpha$ may take values in the multigrading. When $|\alpha|$ is large, the Hilbert-Serre theorem says that $h(\alpha)$ is actually a polynomial function in $\alpha$ and so the generating function is actually a rational function. There are many algorithms to compute the Hilbert function of such rings based on the theory of Gröbner basis and you could try them out in Macaulay2. </p> <p>In certain special cases there are other alternatives though. As in your case, you can often turn this problem into a counting problem. Note that monomials of degree $n$ in $\mathbb{P}_{2,2,2,4}$ correspond bijectively to non-negative solutions of the equation $$2a+2b+2c+4d=n$$Let $s(n)$ denote this number. Then since the ideal $I=(P_1,P_2)$ is a complete intersection of two degree 8 polynomials, we get that the dimension of polynomials of degree $n$ modulo $I$ is exactly $$s(n)-2s(n-8)+s(n-16)$$If $n=18$, we get dimension $125-2\cdot 35+3=60$. In particular, if the ideal is c.i., this argument shows that it suffices to know the number of degree $k$ polynomials in the polynomial ring. In the general case however, you might not be so lucky that your ideal is a c.i. and then perhaps Hilbert-functions are better suited.</p>