This work presents a way to overcome the obstacles of mathematical calculations in obtaining theoretical results. Associated with these rings are several numerical invariants that provide information about the properties of the ring. The intersection properties are a reflection of the combinatorics of monomial exponents, which is often complex.
It is also possible for one group to "contain" another if a subset of the larger group is also a group in its own right. Associativity is already guaranteed by the fact that the operation is associative on the original group G. Having discussed groups, we can now move on to rings, the primary structure we consider in this project. Intuitively, 2 states that (I,+) must be a subgroup of (R,+) and 3 states that multiplying a ring element by an ideal element "raises" the ring element to an ideal.
A classic example of an ideal is the set of even numbers, which under normal addition and multiplication forms an ideal of the group of integers. Given two ideals of a polynomial ring, the ideal with the larger exponent is always a subset of the ideal with the smaller exponent. A corollary to the above is that for two of these ideals, the intersection of the two is the ideal with the larger exponent: (x3) ∩(x2) = (x3).
For example, if R =R[x] and I = (xa), J = (xb), for positive integers a ≥ b, then we can divide the first quadrant of the coordinate plane into two cones.
THE PROBLEM
1, then the algorithm is complete and the Hilbert basis is the set of points in S0. The union of the result of calling this algorithm on all the cones in the plane is the Hilbert set. In general, n extra coordinates will be added to the points of the Hilbert set for the coordinate case in question, where each extra coordinate is found according to the formula.
Now, with the modified Hilbert group G, we can proceed to the second main step of the calculation, which is to find the set of inequalities that constrain a solid in 2 +n dimensions. From these initial inequalities and G, we must derive further inequalities, and the relation of all these inequalities must constrain the solid we are interested in. However, complications arise because, on the one hand, to enter this calculation into a computer, we must know how to write the inequalities explicitly; and on the other hand, sometimes the new inequalities conflict with the original inequalities and these cases require special treatment.
To derive our secondary inequalities, we take each Hilbert set point and replace x −p with x, y − q with y, z1 − r1, etc., and then deny the result, according to the extraInequality function . This is an example of the above case requiring special treatment; the second and fourth subexpressions here, 3x > z and 2y > z, directly contradict our initial inequalities. In these cases it is necessary to discard only those subexpressions that contradict the initial disparities, and keep the others.
After all the inequalities have been found, the last step is to join all the sets of inequalities by connection, giving the boundaries of the solid we are interested in, and integrate and find the volume of this solid. Enter the code in Mathematica to calculate the volume of the polytope defined by the inequalities circled above, which will be the Hilbert-Kunz multiplicity eHK(B). To illustrate the volumes being calculated, we include the 3D objects which are connected to the example a = (3),b = (2).
THE SOLUTION
Using the formula for the determinant of a 2 × 2 matrix, this reduces to finding a solution to the linear equation ua−bv = 1, with the constraint that atu andv must be integers. This is exactly the same as the left side of the equation that we originally solved for finduandv: ua−vb= 1. To implement this part of the program, we simply substitute two generic lists representing the exponents in the ideals, (a1) , a2, a3, . . , an) and (b1, b2, b3, . . . , bn), and a generic point representing the points in G, (p, q, r1, r2, r3, . . . , rn) , in the inequality derivation algorithm to find a general formula for the inequalities that can be derived from each Hilbert set point.
But to avoid writing lists of indefinite length at each step of the calculation, we can start by focusing on a single index, say 1, which gives us only a single a1, b1, r1, and z1 to worry about; further, since we ignore the other ai, bi,ri, and zi, we may dispense with subscripts altogether and simply write a, b, r, and z. We can now substitute the formula derived in this section for the inequalities for each Hilbert set point by adding indices to a, b, z, and r, assuming that . Thus, it is possible to find a general formula for the derivation of the inequality for any number of inputs.
For the parameters, the user only has to enter the two lists of exponents, delimited by spaces, one after the other, without a separator marking the end of one list and the beginning of the next; the list of entered numbers is automatically halved and the first half is assumed to be the exponents of one ideal and the second half the exponents of the other. Otherwise, the program simply displays the values of the Hilbert-Kunz multiplicity and the F signature on two lines and exits. The web interface presents the user with a brief description of the purpose of the website and a text box called n.
This was used to calculate the final values of the integrals giving the Hilbert-Kunz multiplicity and the F-signature. Two conditionals test the validity of the input, and the last multiline command passes all Clozure Common Lisp arguments. The output of the program is valid Mathematica code, allowing its output to be passed to Mathematica through the normal pipeline, and the Sed script removes the Mathematica input and output prompts from the output.
This is used on the server to call the program's command line interface and return the results to the client. The final step of the calculation is an integral which can have any number of dimensions. The other approach is to use a statistical approach to the final integral, sacrificing precision for the sake of computational speed.
You can easily do this by removing the pipeline to Mathematica at the end of your CLI Bash script. Consequently, when ai =bi for all i, Enescu and Spiroff were able to obtain further formulas for the Hilbert-Kunz set.
