Re: [math-fun] Nested radicals considered harmful
Of course, for degree ≥ 5, not all algebraic numbers are expressible in terms of iterated +/-/*/÷ and rational powers of integers. This discussion mentions some interesting related issues, in particular that matrices can be used to express algebraic numbers: <https://math.stackexchange.com/questions/657168/understanding-non-solvable-algebraic-numbers>. —Dan ----- Maybe it's time to look for some other representation of algebraic numbers, instead of nested radicals? Unnested radicals work reasonably well, with near-canonical forms and tolerable computing efforts to manipulate -- I'm thinking +-*/, although / tends to make messes. And strict canonicalization fails because of a factoring problem: if your expression includes sqrt(499999), you won't know it's also 127sqrt(31) without a hint. But if your expression is sqrt(499999) + sqrt(52111), a gcd will provide the hint, without needing to do any factoring. The usefulness of nested radicals for calculating roots of polynomial equations is questionable: We have the cubic and quartic formulas, but using them to get good numerical results requires care: Knuth points out that even the quadratic formula can have numerical problems. Are we better off with a solid numerical routine? The other point of representations, besides numerics, is the benefit of canonical expressions, providing an at-sight equality check, and helping with simplification by allowing visibly equal terms to be combined, and occasionally cancelled. If you plug the quadratic formula back into the polynomial, you can simplify the result down to 0=0. But if you do the same thing with the cubic or quartic formulas, you need to know some special manipulations to get to 0=0. I don't have a good alternative to propose. My only candidate is things like "solve(x^3 = x+1, near 1.3)", which is about as hard to work with as the cubic formula, and offers no obvious scheme for canonicalization. But the present situation is not good. If you need to invoke Gosper or Ramanujan as part of routine simplification of expressions, you've already lost. -----
The answers involving arithmetic modulo a polynomial or companion matrices fail to address the issue of distinguishing between distinct roots of the same polynomial. WFL On 8/29/20, Dan Asimov <dasimov@earthlink.net> wrote:
Of course, for degree ≥ 5, not all algebraic numbers are expressible in terms of iterated +/-/*/÷ and rational powers of integers. This discussion mentions some interesting related issues, in particular that matrices can be used to express algebraic numbers:
<https://math.stackexchange.com/questions/657168/understanding-non-solvable-algebraic-numbers>.
—Dan
----- Maybe it's time to look for some other representation of algebraic numbers, instead of nested radicals?
Unnested radicals work reasonably well, with near-canonical forms and tolerable computing efforts to manipulate -- I'm thinking +-*/, although / tends to make messes. And strict canonicalization fails because of a factoring problem: if your expression includes sqrt(499999), you won't know it's also 127sqrt(31) without a hint. But if your expression is sqrt(499999) + sqrt(52111), a gcd will provide the hint, without needing to do any factoring.
The usefulness of nested radicals for calculating roots of polynomial equations is questionable: We have the cubic and quartic formulas, but using them to get good numerical results requires care: Knuth points out that even the quadratic formula can have numerical problems. Are we better off with a solid numerical routine?
The other point of representations, besides numerics, is the benefit of canonical expressions, providing an at-sight equality check, and helping with simplification by allowing visibly equal terms to be combined, and occasionally cancelled. If you plug the quadratic formula back into the polynomial, you can simplify the result down to 0=0. But if you do the same thing with the cubic or quartic formulas, you need to know some special manipulations to get to 0=0.
I don't have a good alternative to propose. My only candidate is things like "solve(x^3 = x+1, near 1.3)", which is about as hard to work with as the cubic formula, and offers no obvious scheme for canonicalization. But the present situation is not good.
If you need to invoke Gosper or Ramanujan as part of routine simplification of expressions, you've already lost. -----
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It looks like the matrix scheme can't handle situations that involve more than one root of a polynomial. If I've identified the three roots of a three-real-root cubic, say p,q,r with p<q<r, I can write p+2q and distinguish it from q+2p or p+2r etc. I think the matrix method will require determining an extension field where the roots are distinguishable. Rich ------ Quoting Dan Asimov <dasimov@earthlink.net>:
Of course, for degree ? 5, not all algebraic numbers are expressible in terms of iterated +/-/*/÷ and rational powers of integers. This discussion mentions some interesting related issues, in particular that matrices can be used to express algebraic numbers: <https://math.stackexchange.com/questions/657168/understanding-non-solvable-algebraic-numbers>.
?Dan
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