Gives validExponential (EFSTRUX) the correct result?

[IV]

Hallo,

the command "validExponential" from package EFSTRUX (Elementary Function Structure Package) applies Risch's Structure theorem for Elementary functions. The theorem checks if a given exponential or logarithm function is algebraic over a given elementary extension field.

exp(2*log(x)) = x^2 is algebraic over the field \mathbb{C}(x) and algebraic over the field \mathbb{C}(x,e^x). But FriCAS says it isn't:

f:=2*log(x)
g:=exp(f)
K:=kernels([x,exp(x)])
validExponential(K,f,x)

Where I'm wrong?

Is Risch's structure theorem applicable here? Does Risch's structure theorem have an error in this case?

Thanks.

[Waldek Hebisch]

Compare:

(1) -> K := kernels([x,exp(x), log(x)])

x
(1) [x, %e , log(x)]
Type: List(Kernel(Expression(Integer)))
(2) -> validExponential(K, 2*log(x), x)

2
(2) x
Type: Union(Expression(Integer),...)

validExponential assumes that f is in field generated by K, otherwise
it does not work. This assumption is required by Risch structure
theorem. 'validExponential' is a convenience function for
integrator. Logarithm needs somewhat different handling during
integration so there are no similar routine. In fact, internally
FriCAS uses a routine which is very similar to 'validExponential'
but is not exposed to users. Main routine for general use is
'rischNormalize': in general you need to call it before calling
'validExponential', otherwise results may be invalid. Namely
Risch structure theorem requires a field and data structure that
FriCAS uses is guarented to be a field only _after_ running
'rischNormalize'.

--
Waldek Hebisch

[IV]

Am Sonntag, 12. Mai 2019 18:28:15 UTC+2 schrieb Waldek Hebisch:

Oh, I overlooked this constraint. Thank you.

You also answered my other questions. Thank you very much.

And thank you very much for FriCAS and for the Elementary Function Structure Package.