
From:  William Sit 
Subject:  Re: [Axiomdeveloper] Zero divisors in Expression Integer 
Date:  Thu, 04 Jan 2007 01:31:13 0500 
On Thu, 4 Jan 2007 05:47:29 +0100 (CET) Waldek Hebisch <address@hidden> wrote:
I have already written that due to incomplte simplification we may get zero divisors in Expression Integer. Below an easy example that multiplication in Expression Integer is nonassociative(or, if you prefer, a proof that 1 equals 0): (135) > c1 := sqrt(2)*sqrt(3*x)+sqrt(6*x) ++ ++ ++ (135) \6x + \2 \3xType: Expression Integer(136) > c2 := sqrt(2)*sqrt(3*x)sqrt(6*x) ++ ++ ++ (136)  \6x + \2 \3xType: Expression Integer(137) > (1/c1)*c1*c2*(1/c2) (137) 1Type: Expression Integer(138) > (1/c1)*(c1*c2)*(1/c2) (138) 0Type: Expression Integer
But this is not just an Axiom problem. Mathematica does the same thing, with a slight variation on input: a1 = Sqrt[2]*Sqrt[3 Sqrt[5x + 7] + 6]  Sqrt[6Sqrt[5x + 7] + 12] a2 = Sqrt[2]*Sqrt[3 Sqrt[5x + 7] + 6] + Sqrt[6Sqrt[5x + 7] + 12]
(1/a1)*a1*a2*(1/a2) (* answer 1 *)(1/a1)*(a1*a2 // Simplify)*(1/a2) (*answer 0, Simplify is needed to get this *)
The problem seems to be the lack of a canonical form for radical expressions and an algorithm to reduce expressions to canonical form. A related problem is lack of algorithm to test zero. Another is denesting of a nested radical expression. These problems have been studied by Zippel, Landau, Tulone et al, Carette and others.
William
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