description/proof of that for complex polynomial with finite number of variables and their complex conjugates, if polynomial is constantly \(0\), coefficients are \(0\)
Topics
About: field
The table of contents of this article
Starting Context
- The reader knows a definition of polynomials ring over commutative ring.
- The reader admits the proposition that for any polynomial over any field (with more than the-polynomial-degree elements) with any finite number of variables, if the polynomial is constantly \(0\), the coefficients are \(0\).
- The reader admits the proposition that for any linear combination of sines and cosines with any distinct angular velocities, if it is constant, the coefficients are \(0\).
Target Context
- The reader will have a description and a proof of the proposition that for any complex polynomial with any finite number of variables and their complex conjugates, if the polynomial is constantly \(0\), the coefficients are \(0\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(p (x_1, \overline{x_1}, ..., x_n, \overline{x_n})\): \(= \sum_{j_1 + ... + j_n = m} \sum_{s_{1, i} + s_{1, c} = j_1} ... \sum_{s_{n, i} + s_{n, c} = j_n} p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} {x_1}^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}} + ... + \sum_{j_1 + ... + j_n = 0} \sum_{s_{1, i} + s_{1, c} = j_1} ... \sum_{s_{n, i} + s_{n, c} = j_n} p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} {x_1}^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\) where \(j_l, s_{l, i}, s_{l, c} \in \mathbb{N}\), \(\in \{\text{ the polynomials over } \mathbb{C} \text{ with variables, } x_1, \overline{x_1}, ..., x_n, \overline{x_n}\}\)
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Statements:
\(p (x_1, \overline{x_1}, ..., x_n, \overline{x_n}) \equiv 0\)
\(\implies\)
\(\forall l \in \{0, ..., m\} (\forall (j_1, ..., j_n) \text{ such that } j_1 + ... + j_n = l \forall (s_{1, i}, s_{1, c}), ..., (s_{n, i}, s_{n, c}) \text{ such that } s_{1, i} = s_{1, c} = j_1, ..., s_{n, i} = s_{n, c} = j_n ((p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} = 0)))\)
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2: Proof
Whole Strategy: Step 1: regard \(p (x_1, \overline{x_1}, ..., x_n, \overline{x_n})\) as the polynomial of \(x_1, \overline{x_1}\) with each fixed \((x_2, ..., x_n)\), classify the terms according to \(j_1\), and see that the sum of each class is constantly \(0\); Step 2: for each class, see that each coefficient is \(0\); Step 3: regard each coefficient of Step 2 as the polynomial of \(x_2, \overline{x_2}\) with each fixed \((x_3, ..., x_n)\), and see that each coefficient is \(0\); Step 4: and so on.
Step 1:
\(p (x_1, \overline{x_1}, ..., x_n, \overline{x_n}) = (\sum_{s_{1, i} + s_{1, c} = m} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}}) + ... + (\sum_{s_{1, i} + s_{s, c} = 0} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}})\), where \(p_{s_{1, i}, s_{1, c}}\) is a polynomial of \(x_2, \overline{x_2}, ..., x_n, \overline{x_n}\): \(p_{s_{1, i}, s_{1, c}} := \sum_{s_{2, i} + s_{2, c} + ... + s_{n, i} + s_{n, c} = m - (s_{1, i} + s_{1, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}, ..., s_{n, i}, s_{n, c}} {x_2}^{s_{2, i}} {\overline{x_2}}^{s_{2, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}} + ... + \sum_{s_{2, i} + s_{2, c} + ... + s_{n, i} + s_{n, c} = 0 - (s_{1, i} + s_{1, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}, ..., s_{n, i}, s_{n, c}} {x_2}^{s_{2, i}} {\overline{x_2}}^{s_{2, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\): for example, when \(s_{1, i} + s_{1, c} = m\), \(\sum_{s_{2, i} + s_{2, c} + ... + s_{n, i} + s_{n, c} = m - (s_{1, i} + s_{1, c})}\) has only 1 term for \(s_{2, i} + s_{2, c} + ... + s_{n, i} + s_{n, c} = 0\) and \(\sum_{s_{2, i} + s_{2, c} + ... + s_{n, i} + s_{n, c} = 0 - (s_{1, i} + s_{1, c})}\) has \(0\) term.
Let \((x_2, ..., x_n)\) be fixed at any value.
Then, \(p (x_1, \overline{x_1}) := p (x_1, \overline{x_1}, ..., x_n, \overline{x_n})\) is a polynomial of \((x_1, \overline{x_1})\) over \(\mathbb{C}\).
Let \(x_1 = y + z i\).
Then, for each \(s_{1, i} + s_{1, c} = l\), each term of the real part and the imaginary part of \(x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}}\) is an \(l\)-degree-only polynomial of \((y, z)\), and each of the real part and the imaginary part of \(\sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}}\) is \(\sum_{t \in \{0, ..., l\}} c_t y^t z^{l - t}\) for some \(c_t \in \mathbb{C}\) s.
Thinking of each of the real part and the imaginary part of \(p (x_1, \overline{x_1}, ..., x_n, \overline{x_n}) = (\sum_{s_{1, i} + s_{1, c} = m} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}}) + ... + (\sum_{s_{1, i} + s_{s, c} = 0} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}})\), by the proposition that for any polynomial over any field (with more than the-polynomial-degree elements) with any finite number of variables, if the polynomial is constantly \(0\), the coefficients are \(0\), each \(c_t = 0\).
That means that \(\sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} = 0\), constantly with respect to \(x_1\).
In fact, as it holds for each \((x_2, ..., x_n)\), it is constantly \(0\) with respect to \((x_1, ..., x_n)\).
Step 2:
Let us see that for each \(l \in \{0, ..., m\}\) and \(\sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} = 0\), each \(p_{s_{1, i}, s_{1, c}} = 0\).
Let \(x_1 = r e^{\theta i}\).
\(\sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} x_1^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} = \sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} (r e^{\theta i})^{s_{1, i}} (r e^{- \theta i})^{s_{1, c}} = \sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} r^{s_{1, i}} e^{s_{1, i} \theta i} r^{s_{1, c}} e^{- s_{1, c} \theta i} = r^l \sum_{s_{1, i} + s_{1, c} = l} p_{s_{1, i}, s_{1, c}} e^{(s_{1, i} - s_{1, c}) \theta i} = r^l (p_{l, 0} e^{(l - 0) \theta i} + p_{l - 1, 1} e^{(l - 1 - 1) \theta i} + p_{l - 2, 2} e^{(l - 2 - 2) \theta i} + ... + p_{0, l} e^{(0 - l) \theta i}) = r^l (p_{l, 0} e^{l \theta i} + p_{l - 1, 1} e^{(l - 2) \theta i} + p_{l - 2, 2} e^{(l - 4) \theta i} + ... + p_{0, l} e^{- l \theta i}) = r^l ((p_{l, 0} cos (l \theta) + p_{l - 1, 1} cos ((l - 2) \theta) + p_{l - 2, 2} cos ((l - 4) \theta) + ... + p_{0, l} cos (- l \theta)) + (p_{l, 0} sin (l \theta) + p_{l - 1, 1} sin ((l - 2) \theta) + p_{l - 2, 2} sin ((l - 4) \theta) + ... + p_{0, l} sin (- l \theta)) i) = r^l ((p_{l, 0} cos (l \theta) + p_{l - 1, 1} cos ((l - 2) \theta) + p_{l - 2, 2} cos ((l - 4) \theta) + ... + p_{0, l} cos (l \theta)) + (p_{l, 0} sin (l \theta) + p_{l - 1, 1} sin ((l - 2) \theta) + p_{l - 2, 2} sin ((l - 4) \theta) + ... - p_{0, l} sin (l \theta)) i) \equiv 0\).
Looking at the real part, \(p_{l, 0} cos (l \theta) + p_{l - 1, 1} cos ((l - 2) \theta) + p_{l - 2, 2} cos ((l - 4) \theta) + ... + p_{0, l} cos (l \theta) = (p_{l, 0} + p_{0, l}) cos (l \theta) + (p_{l - 1, 1} + p_{1, l - 1}) cos ((l - 2) \theta) + (p_{l - 2, 2} + p_{2, l - 2}) cos ((l - 4) \theta) + ...\), where the last term is \(p_{l / 2, l / 2} cos (0 \theta) = p_{l / 2, l / 2}\) when \(l\) is even and \((p_{(l + 1) / 2, (l - 1) / 2} + p_{(l - 1) / 2, (l + 1) / 2}) cos (\theta)\) when \(l\) is odd.
By the proposition that for any linear combination of sines and cosines with any distinct angular velocities, if it is constant, the coefficients are \(0\), \(p_{l, 0} + p_{0, l} = p_{l - 1, 1} + p_{l - 1, 1} = ... = p_{l / 2, l / 2} \text{ or } p_{(l + 1) / 2, (l - 1) / 2} + p_{(l - 1) / 2, (l + 1) / 2} = 0\): \(\{l, l - 2, ..., 0 \text{ or } 1\}\) is distinct, while the \(0\) term (if exists) becomes the constant.
Looking at the imaginary part, \(p_{l, 0} sin (l \theta) + p_{l - 1, 1} sin ((l - 2) \theta) + p_{l - 2, 2} sin ((l - 4) \theta) + ... - p_{0, l} sin (l \theta) = (p_{l, 0} - p_{0, l}) sin (l \theta) + (p_{l - 1, 1} - p_{1, l - 1}) sin ((l - 2) \theta) + (p_{l - 2, 2} - p_{2, l - 2}) sin ((l - 4) \theta) + ...\), where the last term is \(p_{l / 2, l / 2} sin (0 \theta) = 0\) when \(l\) is even and \((p_{(l + 1) / 2, (l - 1) / 2} - p_{(l - 1) / 2, (l + 1) / 2}) sin (\theta)\) when \(l\) is odd.
Likewise, \(p_{l, 0} - p_{0, l} = p_{l - 1, 1} - p_{l - 1, 1} = ... = 0 \text{ or } p_{(l + 1) / 2, (l - 1) / 2} - p_{(l - 1) / 2, (l + 1) / 2} = 0\).
That means that \(p_{l, 0} = p_{l - 1, 1} = ... p_{0, l} = 0\).
As that holds for each \((x_2, ..., x_n)\), \(p_{l, 0} = p_{l - 1, 1} = ... p_{0, l} \equiv 0\) with respect to \((x_2, ..., x_n)\).
Step 3:
Each \(p_{s_{1, i}, s_{1, c}} (x_2, \overline{x_2}, ..., x_n, \overline{x_n})\) is a polynomial of \((x_2, \overline{x_2}, ..., x_n, \overline{x_n})\) constantly \(0\).
\(p_{s_{1, i}, s_{1, c}} (x_2, \overline{x_2}, ..., x_n, \overline{x_n}) = (\sum_{s_{2, i} + s_{2, c} = m - (s_{1, i} + s_{1, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}} x_2^{s_{2, i}} {\overline{x_2}}^{s_{2, c}}) + ... + (\sum_{s_{2, i} + s_{2, c} = 0 - (s_{1, i} + s_{1, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}} x_2^{s_{2, i}} {\overline{x_2}}^{s_{2, c}})\), where \(p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}}\) is a polynomial of \(x_3, \overline{x_3}, ..., x_n, \overline{x_n}\): \(p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}} := \sum_{s_{3, i} + s_{3, c} + ... + s_{n, i} + s_{n, c} = m - (s_{1, i} + s_{1, c} + s_{2, i} + s_{2, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}, s_{3, i}, s_{3, c}, ..., s_{n, i}, s_{n, c}} {x_3}^{s_{3, i}} {\overline{x_3}}^{s_{3, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}} + ... + \sum_{s_{3, i} + s_{3, c} + ... + s_{n, i} + s_{n, c} = 0 - (s_{1, i} + s_{1, c} + s_{2, i} + s_{2, c})} p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}, s_{3, i}, s_{3, c}, ..., s_{n, i}, s_{n, c}} {x_3}^{s_{3, i}} {\overline{x_3}}^{s_{3, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\): for example, when \(s_{1, i} + s_{1, c} + s_{2, i} + s_{2, c} = m\), \(\sum_{s_{3, i} + s_{3, c} + ... + s_{n, i} + s_{n, c} = m - (s_{1, i} + s_{1, c} + s_{2, i} + s_{2, c})}\) has only 1 term for \(s_{3, i} + s_{3, c} + ... + s_{n, i} + s_{n, c} = 0\) and \(\sum_{s_{3, i} + s_{3, c} + ... + s_{n, i} + s_{n, c} = 0 - (s_{1, i} + s_{1, c} + s_{2, i} + s_{2, c})}\) has \(0\) term.
Let \((x_3, ..., x_n)\) be fixed at any value.
Then, \(p_{s_{1, i}, s_{1, c}} (x_2, \overline{x_2}) := p_{s_{1, i}, s_{1, c}} (x_2, \overline{x_2}, ..., x_n, \overline{x_n})\) is a polynomial of \((x_2, \overline{x_2})\) over \(\mathbb{C}\).
By the logic parallel to Step 2, \(p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}} \equiv 0\) constantly with respect to \((x_3, ..., x_n)\).
Step 4:
And so on, after all, we think of the polynomials of \((x_n, \overline{x_n})\), whose coefficients are constants, and the coefficients are \(0\).
But each \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}}\) is one of those constant coefficients, because \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} {x_1}^{s_{1, i}} {\overline{x_1}}^{s_{1, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\) is put into \(p_{s_{1, i}, s_{1, c}}\) as \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} {x_2}^{s_{2, i}} {\overline{x_2}}^{s_{2, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\), which is put into \(p_{s_{1, i}, s_{1, c}, s_{2, i}, s_{2, c}}\) as \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} {x_3}^{s_{3, i}} {\overline{x_3}}^{s_{3, c}} ... {x_n}^{s_{n, i}} {\overline{x_n}}^{s_{n, c}}\), ..., which becomes \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}}\) as a constant coefficient of \(p_{s_{1, i}, s_{1, c}, ..., s_{n - 1, i}, s_{n - 1, c}}\).
So, \(p_{s_{1, i}, s_{1, c}, ..., s_{n, i}, s_{n, c}} = 0\).
