 This algebraic geometry video will be a review of Hilbert polynomials of graded modules over a graded ring. So suppose m is a graded module over a graded ring where this ring will be generated over a field k by various elements x1 up to x k where the degree of xi is going to be some number di greater than zero. So it will be a question of this by some graded ideal of course. So for applications we might take r to be the same as m and r might be the graded ring associated with some projective variety. So if you've got a projective variety generated by some, sorry corresponding to some graded ideal, then this gives us a graded ring and we can ask what is the growth rate of these components mn and the growth rate of the mn might tell us something about the variety. If these numbers grow big it suggests the variety is big in some sense. So we encode these dimensions as the following function fmx and we're just going to encode it as sum of x to the n times the dimension of mn where this is the dimension of mn as a vector space over the field k and we will take m to be finitely generated and we want to know what does this function look like and the basic result is that fmx is a rational function. Moreover it's a rational function with a very restricted denominator as we will see in a moment. And what you do is you just look at the following exact sequence you have nought goes to the kernel of xk so we pick one of the generators goes to m goes to mdk goes to mdk over xk times m. So here this map here is multiplication by xk and we have this exact sequence and you can now look at the graded pieces of each of them and if you've got an exact sequence of vector spaces then the dimension of this minus the dimension of this plus the dimension of this minus the dimension of this is always equal to zero so we get a relation between the functions of these four modules. Moreover this module here is a module over our quotient out by xk and this thing here is also a module over the ring r quotient out by xk. So f of mdk is equal to x the dk times fm I forgot to say m of dk means m with the graded shifted by dk so this just means shift the grading. So we see one minus x the minus dk times fm is something involving this module and this module and these are modules over rings with smaller numbers of generators so by induction these are rational functions. So this shows that f of m is a rational function. So what we find is that f I guess I was actually writing f of m where I should have written subscript. So f of m is a rational function with denominator dividing one minus x the d1 one minus x to the d2 and so on one minus x to the dk gained by induction on k. Well the most important special case is when all the di are equal to one although it's sometimes useful to have this more general version. So for instance this would just happen if you took the polynomial ring on generators x1 to xn and gave everything the obvious grading. In this case the denominator is one minus x to the power of k and if we expand one over one minus x to the k we see that it's sorry the k should be there. We see that the coefficients of the powers of x for i greater than or equal to naught are given by polynomial in i for instance one over one minus x to the one is one plus x plus x squared and so on one over one minus x squared is one plus two x plus three x squared and so on and in general the coefficients here are actually given by certain binomial coefficients which are going to be polynomials so their degree naught for one minus x to the one degree one for one minus x squared and so on. Notice that they're only a polynomial i for i greater than or equal to zero for i less than zero of course they're well they're still given by polynomial but they're given by different polynomials so there's a sort of discontinuity in the behavior of these coefficients at x to the power of zero so what this means is the dimension of m n the degree n piece of the module m is a polynomial in n for n large and greater than zero of course and if n is sufficiently small then you run into the problem that these things here are not polynomials for x negative sorry for negative powers of x so this is called the Hilbert polynomial of of m so the Hilbert polynomial describes how fast the graded pieces of m grow in large degree well this polynomial has the following rather special property in general it's got um in in in general its coefficients need not be integers however this polynomial has the following property that f of n is an integer for n an integer so this is an integer valued polynomial that's because for any number n f of n is always going to be the dimension of some graded piece of m well actually that's true for n sufficiently large but if if a polynomial is integer for large integer values then it's integer for all values so um what do integer valued polynomials look like well pretty obviously if a naught if all the numbers a i are zero then a naught plus a one x plus plus a k x to the k is always an integer if x is an integer and all the a is integers and you can ask the converse if you've got an integer valued polynomial are all its coefficients integers we're obviously not because x times x minus one over two is always an integer for x an integer but its coefficients are x squared over two minus x over two so an integer valued polynomial need not have integer coefficients so what we want to do now is to classify the integer valued polynomials well let's let's write down some integer valued polynomials well there are some obvious examples you can take one or x or x times x minus one over two or x times x minus one times x minus two over three factorial and you notice these are all just binomial coefficients so this is x choose zero x choose one x choose two x choose three and so on and binomial coefficients are always integers so these are certainly integer value polynomials and what we're going to show is that these actually span all integer valued polynomials and one way to see that is to look at the values of these for small integers so that the value for x being zero one two three four in some of these polynomials is as follows first of all this one is one for x equals one and we don't really care what it is for other values of x the next one is zero for x equals zero and one for x equals one and we don't care what the other values are this one is zero for x equals zero or one and it's one for x equals two and we don't care elsewhere and this one is zero for the first three values of x and so on so we're sort of getting all we're interested in are these values and we don't care what they are anywhere else now I suppose f of n is a degree k integer valued polynomial and now we can arrange for a linear combination so we can find a linear combination of these binomial polynomials which is zero for x equals zero one up to k and that's because um we can first choose a zero to make it vanish at x equals one and then we can choose a one to make it vanish sorry we choose a zero to make it vanish at x equals zero then we choose a one to make it vanish at x equals one and since these numbers are zero here that won't affect the value at x equals zero and similarly we choose a two to make it vanish at x equals two and and so on, so we can get up to this. And now this is a polynomial of degree K vanishing at K plus one points, naught up to K. So it is zero for all K, for all N. As a polynomial degree K can't have K plus one zeros. So this shows that any integer-valued polynomial such as a Hilbert polynomial must be a linear combination of these polynomials. A particularly important special case is that the leading coefficient of F is of the form A K X for K over K factorial where A K is an integer. So although the coefficient of X for K may not be an integer, the coefficient of X for K when multiplied by K factorial is always an integer. And this integer will turn out to give us the degree of various algebraic varieties. There's a useful variation of the Hilbert polynomial. Instead of looking at the dimension of M N, all we actually used about the dimension was additive in short exact sequences. And we can use any other measure of the size of M N provided it's additive on short exact sequence and get a sort of Hilbert polynomial out of that. So next lecture we'll describe some applications