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I have an expression where the variables are algebraic integers: $p4 = \frac{p12 - p41 \cdot p21}{p22}$ p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and p21 is degree 24. I am trying to avoid a geometric increase in degrees, p41*p21 is 32*24 = 768 and p12 - p41*p21 would be 48*768 = 36864 and then dividing this by p22 would give 48*36864 = degree 1769472 would is well beyond the capability of my computer (using GP-Pari)

How can this be simpified? I strongly suspect that p42 is degree 192, but have no way to determine this unless I grind it out using the algdep command of GP-Pari which depends upon the accuracy of the digits (which will have to be around 20K) This would take several days to run.

Thanks for your suggestions.

  • Randall
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  • $\begingroup$ Perhaps you have more info that could reduce the degree. For example, the fields generated individually by $p_{41}$ and $p_{21}$ could intersect in a field of degree 16. and similar things.Also possibly a a convenient automorphism (a Galois group element) might send your number $p_4$ to a number more amenable for computations. $\endgroup$
    – P Vanchinathan
    Commented May 14, 2013 at 6:19
  • $\begingroup$ What is p42? Is that the same as p4 in the first paragraph? $\endgroup$
    – Harm Derksen
    Commented May 14, 2013 at 15:16
  • $\begingroup$ @Randall: Isn't your question essentially the same one as your question mathoverflow.net/questions/24513/… from 3 years ago? How are your numbers $pxy$ given, by minimal polynomials or complex floats up to some precision? $\endgroup$
    – Peter Mueller
    Commented May 14, 2013 at 16:05

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