加洛瓦定理是否存在复杂性观点？

• 伽罗瓦定理有效地表明，不能使用系数和根的有理函数来表达> = 5的多项式的根-这难道不是说给定多项式没有确定性算法可找到根吗？

• 现在考虑以下形式的决策问题：“给定实根多项式$p$且数字k 至少在间隔k处是的第三和第四高根$p$？”

该决策问题的证明证书仅是该多项式的根的集合，即简短证书，因此看起来$NP$ BUT不是Galois定理，它说不存在任何确定性算法来为此找到证书决定问题？（如果为true，则此属性排除任何算法来决定该问题的答案）

那么，这个决策问题位于哪一类复杂性中？

我见过的所有NP完全问题总是有一个简单的指数时间算法可以解决。我不知道这是否应为对所有NP完整问题都应该正确的属性。对于这个决策问题，这似乎并非正确。

有趣的联系是，但是Galois理论指出，不存在（一致的）使用根基查找五分力的根的方法，而不是说问题具有解决方案（例如，最长的路径），可能需要超多项式时间。所以我想说这与不确定性而不是复杂性有关。

具体来说，在伽罗瓦理论中，人们逐步地建立方程根的群扩展（一次添加一个根）。从某种意义上说，所有这些组都应该是可解的，在以其他顺序构建这些扩展的过程中应该没有歧义。关于MO关于构造方程的Galois群的复杂性存在一个相关问题。

此处的另一个参考文献：“计算盖洛伊理论：不变性和计算”，克劳斯·菲克·于尔根·克鲁纳斯$\mathbb{Q}$

此外，基于方程的Galois组的构造，可以用根系统地表示多项式等式的根（当方程可使用根求解时）。参考：“多项式根的根表示”，Hirokazu Anai Kazuhiro横山，2002年

确定给定的整数一元不可约多项式是否可被基团溶解的计算复杂度在P Ref “自由基的可溶性在多项式时间内”，S。Landau GL Miller 1984$\mathbb{Z}$$\mathbb{P}$

亚历山大·赫尔普克（Alexander Hulpke）最近对“伽罗瓦群计算技术”的调查

当然，如果人们正在寻找好的近似算法及其复杂性（例如，牛顿法或Sturm定理），那么这是一个稍有不同的问题，已经发布的答案提供了该方向的更多信息。

我假设您正在考虑使用整数系数的多项式。

您为调查选择了错误的起点；您的目标是找到真实根的良好估计。寻找代数公式以便您可以对其进行足够精确的评估是可以做的事情，但这并不是正确的选择。（当然，除非“ k多项式的最大实根”是您的代数运算之一）

一个更好的起点是使用Sturm定理来分离多项式的根。然后，您可以通过二进制搜索生成更好的估计值，但是如果这样做太慢，则可以使用牛顿方法快速生成高精度的估计值。

但这仅仅是寻找证书。仍然存在哪些证书的问题。

首先，我要指出的是，您可以直接计算两个根的相距是否恰好是 单位，例如，通过计算gcd （p （x ），p （x - k ））。您还必须决定要对重复的根进行处理并进行适当处理。我认为您将特别处理这些情况。$k$$\gcd(p(x), p(x-k))$

如果我们知道两个根之间的距离不完全是单位，则意味着您可以得出足够精确的估计值，以证明它们大于或小于k个单位。例如，有两种证书：$k$$k$

The first kind (proof in the negative) is

• $a$ is not a root of $p$
• $p$ has no roots in $(a-k, a)$
• $p$ has three roots in $(a, \infty)$

The second kind (proof in the positive) is

• $a$ is not a root of $p$
• $p$ has at least two roots in $(a-k,a)$
• $p$ has two roots in $(a, \infty)$

A certificate can be verified by using Sturm's theorem. Now, your question about the size of a certificate boils down to finding how many bits of precision you need to represent $a$.

In other words, what are the bounds on the possible values of $a-b-k$, where $a,b$ are roots of $f$?

I'm not sure of a great approach, but one that should give you something is to observe that all of these values are roots of the polynomial:

Why? Recall that the resultant of two monic polynomials is the product of all differences of their roots, so

where $c$ is the leading coefficient and $d$ is the degree of $f$. (maybe I've written the formula for $-g(x)$ instead of $g(x)$; I'm never sure on the sign)

So the question is to find estimates for how large the coefficients $g$ can be, and then once you know that, find estimates to how close a root of $g$ can be to zero.

(or, alternatively, find the largest magnitude that a root of the reverse polynomial of $g$ can have; the roots of the reverse polynomial are the inverses of the roots of $g$)

am going to take your questions as mostly open ended. the galois proof now known as the Abel-Ruffini thm shows the impossibility of polynomial solutions to the quintic. (in contrast to eg the quadratic equation). so its not really a result on the hardness of a problem per se but rather the impossibility. in this sense it is more analogous to eg a proof of undecidability of the halting problem. complexity theory is in general concerned with the "cost" of computing solutions. that is the viewpoint of two leading CS researchers in the introductory section of this following paper (Computability and Complexity / Kleinberg & Papadimitriou), sec 1 The Quest for the Quintic Formula:

Viewed from the safe distance of a few centuries, the story is clearly one about com- putation, and it contains many of the key ingredients that arise in later efforts to model computation: We take a computational process that we understand intuitively (solving an equation, in this case), formulate a precise model, and from the model derive some highly unexpected consequences about the computational power of the process. It is precisely this approach that we wish to apply to computation in general.

elsewhere a loose/ general analogy might be that a P$\neq$NP proof (or other complexity class separation) is analogous to a computational impossibility result somewhat like the Abel-Ruffini thm. a separation result says roughly that problems of a certain type cannot be solved with "computational resources" of another certain type. a P$\neq$NP theorem would be viewed as a (monumental) computational impossibility result.