Space conics

Jonathan Hauenstein and Frank Sottile
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    We consider the enumerative geometry problem of counting the number of real space conics that meet k points and 8-2k lines for k = 0, 1, 2. First, for each k, we solved a random complex instance using Bertini. We then solved many real instances with Bertini using a parameter homotopy and certifed the number of real solutions using alphaCertified.
    The remainder of this page documents how to perform these calculations and the files we used and created in the course of this test.
Directions
Before you begin, you will need to have a working binary of Bertini and alphaCertified on your machine along with the GMP library.
Note that we describe this computation for k = 2 and provide the necessary files for k = 0 and k = 1 at the bottom of the page. Assuming that we have the start points and created Instances2.out, the following bash session computes and certifies the solutions for a random real instance.
>: ./Instances2.out 
>: bertini Conic2.bertini 
>: alphaCertified Conic2.poly nonsingular_solutions settings > output 
>: sh scour.sh 
Frequency table for 500 instances verified with alphaCertified.
# real  024 total
frequency  12 221 267 500

Files for other values of k
For k = 1:
Random complex instance: Conic1orig.bertini.
Start points for parameter homotopy: start.
Code to create files for a random real instance: Instances1.c.
Frequency table for 1,000,000 instances verified with alphaCertified.
# real  0246 8101214 1618total
frequency  0 3,295 21,764  89,350 193,172 261,046  227,018 137,990 52,865  13,500 1,000,000
40,000 instances were also tested symbolically, with a perl script that ran Singular to compute an eliminant and ran Maple to determine the number of real roots of that eliminant. We display the frequency table. Here is a .tar.gz file that creates the files used in this computation.
# real  0246 8101214 1618total
frequency  0 146  892 3558  7739 10575  8965 5488  2089  548 40000
For k = 0:
Random complex instance: Conic0orig.bertini.
Start points for parameter homotopy: start.
Code to create files for a random real instance: Instances0.c.
Frequency table for 15,662,000 instances verified with alphaCertified.
# real0246 810121416 1820222426 2830323436 3840424446 4850525456 5860626466 6870727476 7880828486 889092total
frequency182665 4661,5484,76511,928 26,43952,87598,129167,932 270,267404,918569,891,316756,527 942,6741,114,0331,246,5331,332,289 1,355,3201,319,6991,226,6671,091,019 932,838762,463596,174449,021 323,927223,455149,62995,740 59,14134,83419,51610,672 5,6712,7441,290530 2049026113 2015,662,000

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Last modified: Wed Nov 3 17:18:22 CDT 2010