# In this Maple sheet we examine the algebraic complexity of the
# SideOfOctagon predicate, which can be evaluated with two calls
# to the InCircle predicate, as described in the paper. The input to
# the predicate is a rational point LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw== and we check on which side of
# the circles LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJ0YyRjUtRiM2JS1GLzYlUSJFRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHRj0vJS9zdWJzY3JpcHRzaGlmdEdGPS9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+ and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUtRiM2JS1GLzYlUSJFRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGRy9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+ is located LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEicEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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, is
# defined by the points 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 and 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5 is intended in
# modulo 8.
# Initially, we report the coordinates of the necessary points in
# Do here. We have converted them in expressions of \316\276. JSFHQyQ+Jkkidkc2IjYjIiIhLUkncnRhYmxlRyUqcHJvdGVjdGVkRzYmOyIiIiIiIztGLkYuNyQ3IywkKiZGLyNGLiIiJUkjeGlHRiZGLiNGLkYvNyMsJComRi9GNUY3ISIiI0Y8Ri8vSShzdWJ0eXBlRzYkRitJKF9zeXNsaWJHRiZJJ01hdHJpeEdGQEY8QyQ+Jkkidkc2IjYjIiIiLUkncnRhYmxlRyUqcHJvdGVjdGVkRzYmO0YoIiIjO0YoRig3JDcjLCQqJkYuI0YoIiIlLCYqJi1JJXNxcnRHNiRGK0koX3N5c2xpYkdGJjYjRi5GKEkjeGlHRiZGKEYoKiZGOEYoRj0hIiJGKEYoRjQ3IywkKiZGLkY0LCZGN0YoRj5GP0YoRjQvSShzdWJ0eXBlR0Y6SSdNYXRyaXhHRjpGPw==QyQ+Jkkidkc2IjYjIiIjLUkncnRhYmxlRyUqcHJvdGVjdGVkRzYmOyIiIkYoO0YuRi43JDcjLCQqJkYoI0YuIiIlSSN4aUdGJiEiIiNGLkYoNyMsJComRihGNEY2Ri5GOC9JKHN1YnR5cGVHNiRGK0koX3N5c2xpYkdGJkknTWF0cml4R0Y+Rjc=QyQ+JkkibUc2IjYjIiIhLUkncnRhYmxlRyUqcHJvdGVjdGVkRzYmOyIiIiIiIztGLkYuNyQ3IyokSSN4aUdGJiEiIjcjRigvSShzdWJ0eXBlRzYkRitJKF9zeXNsaWJHRiZJJ01hdHJpeEdGOUY1QyQ+JkkibUc2IjYjIiIiLUkncnRhYmxlRyUqcHJvdGVjdGVkRzYmO0YoIiIjO0YoRig3JDcjLCQqJi1JJXNxcnRHRiY2I0YuRihJI3hpR0YmISIiI0YoRi5GMS9JKHN1YnR5cGVHRiZJJ01hdHJpeEc2JEYrSShfc3lzbGliR0YmRjg=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QyQ+SSdvcmllbnRHNiJmKjYlSSJwR0YlSSJxR0YlSSJyR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtJkkuTGluZWFyQWxnZWJyYUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSxEZXRlcm1pbmFudEdGJTYjLUknTWF0cml4R0YxNiM3JTclJjkkNiQiIiJGPyZGPTYkIiIjRj9GPzclJjklRj4mRkVGQUY/NyUmOSZGPiZGSUZBRj9GJUYlRiVGPw==PkkpaW5jaXJjbGVHNiJmKjYmSSJwR0YkSSJxR0YkSSJyR0YkSSJzR0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQsKiomLCYqJCY5JDYkIiIiRjUiIiNGNSokJkYzNiRGNkY1RjZGNUY1LUknb3JpZW50R0YkNiU5JTkmOSdGNUY1KiYsJiokJkY9RjRGNkY1KiQmRj1GOUY2RjVGNS1GOzYlRjNGPkY/RjUhIiIqJiwmKiQmRj5GNEY2RjUqJCZGPkY5RjZGNUY1LUY7NiVGM0Y9Rj9GNUY1KiYsJiokJkY/RjRGNkY1KiQmRj9GOUY2RjVGNS1GOzYlRjNGPUY+RjVGSEYkRiRGJA==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JSFH# We compute the InCircle predicate for 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PkkkZXgxRzYiLUkpc2ltcGxpZnlHRiQ2Iy1JJ2V4cGFuZEclKnByb3RlY3RlZEc2Iy1JKWluY2lyY2xlR0YkNiYmSSJ2R0YkNiMiIiEmSSJtR0YkRjEmRjA2IyIiIkkicEdGJA==# The denominator of ex1 is clearly positive, so we examine the numeratorLUkpc3BsaXRhbGxHNiI2Iy1JJm51bWVyRyUqcHJvdGVjdGVkRzYjSSRleDFHRiQ=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=# Now we examine the complexity of InCircle for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobXN1YnN1cEdGJDYoLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0YyRjUtRiM2JS1GLzYlUSJFRidGMkY1RjJGNS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJS9zdWJzY3JpcHRzaGlmdEdGRy9JK21zZW1hbnRpY3NHRiRRLFtub25lLG5vbmVdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJ0Y+.PkkkZXgyRzYiLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2Iy1JJ2V4cGFuZEdGKDYjLUkpaW5jaXJjbGVHRiQ2JiZJInZHRiQ2IyIiIiZJIm1HRiRGMyZGMjYjIiIjSSJwR0Yk# Here again the denominator is positive, so we examine the numerator LUkpc3BsaXRhbGxHNiI2I0kkZXgyR0YkLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=# Since the maximum degree of the quantities LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiR0YnRi9GMkY1LUYsNiVRIkhGJ0YvRjJGNS1GLDYlUSJKRidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5 is 2 in both cases,
# and due to \316\276, we conclude that the degree of the predicate SideOfOctagon
# is 8 in the input coordinates.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=