InequalitySolve with algebraic numbers and Simplify

• To: mathgroup at smc.vnet.net
• Subject: [mg22987] InequalitySolve with algebraic numbers and Simplify
• From: Gianluca Gorni <gorni at dimi.uniud.it>
• Date: Sun, 9 Apr 2000 01:45:52 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Hello!

I had a system of *linear* inequalities in two variables x,y,
with simple algebraic numbers as coefficients,
to solve with InequalitySolve[], and I naively assumed
that the solution would be a set of likewise *linear*
inequalities. So I was surprised at results like this:

Needs["Algebra`InequalitySolve`"]

InequalitySolve[{y <= x*Sqrt[2], y <= x}, {x, y}]

x <= 0 && y <= -(Sqrt[2]*Sqrt[x^2]) || x > 0 && y <= x

Strictly speaking the result is correct, but it does not look
good, because of that Sqrt[x^2]. I tried with
Experimental`CylindricalAlgebraicDecomposition,
but it gives exactly the same answer.

Trying to reduce the results to a more manageable form, I
met some somewhat disappointing behaviour of Simplify[]
with assumptions (of Mathematica version 4):

In:   Simplify[Sqrt[x^2], x == 1 + Sqrt[2]]

Out:  Sqrt[x^2]

It seems that Mathematica doesn't notice that 1+Sqrt[2] is real and
positive! So let's teach Mathematica that it is real, at least:

In:  Simplify[Sqrt[x^2], {Element[x, Reals], x == 1 + Sqrt[2]}]

Out: x

Somehow I would have expected the answer to be 1+Sqrt[2], what
about you? Even if x has a smaller LeafCount than 1+Sqrt[2].

Best regards,

Gianluca Gorni

--

+---------------------------------+
| Gianluca Gorni                  |
| Universita` di Udine            |
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|   e Informatica                 |
| via delle Scienze 208           |
| I-33100 Udine UD                |
| Italy                           |
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```

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