Edit detail for #285 equal in AlgebraicNumber fails revision 2 of 3

FriCAS? now modified equality so the problem is not present.

I try :

a := 3 + sqrt 5

(1)

(-a = (a^2)^(1/2))::Boolean

(2)

But

(sqrt 2 = - sqrt 2)::Boolean

(3)

I expect the same result, and I prefer false.

Do you consider to be a particular (single) solution of , e.g. the positive one, or the general solution of this polynomial? I prefer the latter, so I think both (2) and (3) should be true.

Axiom's sqrt does not seem to be multiple valued, so therefore I disagree with you Bill, it cannot be the general solution of the polynomial. As with most other systems, sqrt is taken to be the positive root only, and if it comes from the solution of a quadratic, the +/- multiple valued-ness must be encoded elsewhere.

e.g.

sqrt(2)::Float

(4)

gives only one value.

Also note that axiom is not even self-consistent here:

-a::Float

(5)

(a^2)^(1/2)::Float

(6)

which clearly are not equal.

If you write:

-a=(a^2)^(1/2)

(7)

coerce(%%(7))$Equation(AlgebraicNumber)

(8)

you are asking about equality in the domain of AlgebraicNumber? (whatever that happens to be). But if you write:

coerce(%%(7)::Equation(Float))$Equation(Float)

(9)

then you are referring to equality in the domain Float. These need not be the same thing. But I agree that Axiom does not have an entirely consistent treatment of sqrt even within a given domain. :(

From:

http://wiki.axiom-developer.org/axiom--test--1/src/algebra/ConstantSpad

In AlgebraicNumber? we find:

    a=b == trueEqual((a-b)::Rep,0::Rep)

In InnerAlgebraicNumber? it says:

    trueEqual(a,b) ==
       -- if two algebraic numbers have the same norm (after deleting repeated
       -- roots, then they are certainly conjugates. Note that we start with a
       -- monic polynomial, so don't have to check for constant factors.
       -- this will be fooled by sqrt(2) and -sqrt(2), but the = in
       -- AlgebraicNumber knows what to do about this.

but unfortunately is seems that the special treatment of sqrt(2) (and similar algebraic numbers) was never implemented. :(

equality in InnerAlgebraicNumber? --Bill Page, Wed, 19 Apr 2006 20:35:13 -0500 reply

InnerAlgebraicNumber? (IAN) is described as

Algebraic closure of the rational numbers.

while AlgebraicNumber? (AN) is described as

Algebraic closure of the rational numbers, with mathematical =

Apparently IAN implements trueEqual as the equivalence relation defined by conjugation:

  trueEqual : (%,%) -> Boolean
      ++ trueEqual(x,y) tries to determine if the two numbers are equal

trueEqual(sqrt(2),-sqrt(2))$IAN

(10)

This is quite strange. As I understand it, AN is almost IAN (only three functions have over-riding implementations: zero? , one? , and = ) and yet, = in AN is not transitive, but apparently, trueEqual in IAN is:

a:AN:=sqrt(5)+3

(11)

b:=(a^2)^(1/2)

(12)

(a=b)::Boolean

(13)

(a=-b)::Boolean

(14)

(b=-b)::Boolean

(15)

f:IAN:=sqrt(5)+3

(16)

g:=(f^2)^(1/2)

(17)

trueEqual(f,-g)

(18)

trueEqual(f,g)

(19)

trueEqual(g,-g)

(20)

But this is deceptive since trueEqual(g, -g) is not trueEqual(g+g,0). Note we cannot use trueEqual(a,b) or trueEqual(a-b,0) since there is no coercion from AN to IAN . Moreover, trueEqual is not transitive either.

fricas

trueEqual(g+g,0)

not the same as trueEqual(g, -g)

(21)

Type: Boolean

fricas

trueEqual(-f,g)

(22)

Type: Boolean

fricas

trueEqual(f,-f)

(23)

Type: Boolean

There is also = in IAN, which is implemented using = in EXPR INT .

Perhaps trueEqual should be called conjugate? instead and exported from AN, and = in AN is really not possible. Also, trueEqual is only used in constant.spad and changing its name will not affect any other constructors. But

http://mathworld.wolfram.com/ConjugateElements.html

Two elements alpha, beta of a field K, which is an extension field of a field F, are called conjugate (over F) if they are both algebraic over F and have the same minimal polynomial. ... This conjugacy relation is an equivalence relation on the set of algebraic elements in a given extension K of the field F.

So if trueEqual is supposed to implement conjugate? then the bug must be in trueEqual, I think.

Some details of the algorithm are here:

http://www.csd.uwo.ca/~watt/pub/reprints/1995-issac-dynev.pdf

See especially reference [DDD].

Anonymous wrote:

I expect the same result, and I prefer false.

This result is possible using the RealClosure? package:

Ran := RealClosure( Fraction(Integer) )

(24)

a1:Ran := 3 + sqrt 5

(25)

(-a1 = (a1^2)^(1/2))::Boolean

(26)

a2:=sqrt(2)$Ran

(27)

a3:=-sqrt(2)$Ran

(28)

(a2=a3)::Boolean

(29)