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Can someone please explain (with an example or two) what is meant by “a polynomial $p(x_1, ..., x_n)$”? Stillwell starts referring to them without defining them and they are beyond my elementary mathematics education. He usually adds “with integer coefficients.”

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  • $\begingroup$ To make it simple, it is a sum of monomials in the indeterminates $x_1,x_2,\dots x_n$. For instance: $p(x,y,z)=5x^2y-3xyz+2z^4+1$ is a polynomial in the ring of polynomials $\mathbf Z[x, y,z]$. $\endgroup$
    – Bernard
    Commented Jan 30, 2020 at 9:15
  • $\begingroup$ @DietrichBurde that looks similar to what my searches turned up. Was hoping for something not in abstract algebra terms. $\endgroup$ Commented Jan 30, 2020 at 9:20
  • $\begingroup$ Reading “Elements of Mathematics” by John Stillwell. I have an engineering background and am a high school mathematics teacher. So hoping for answers that don’t jump to formal definitions from abstract algebra. $\endgroup$ Commented Jan 30, 2020 at 9:29

3 Answers 3

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A polynomial in $x_1,x_2,..,x_n$ is a finite sum of terms of the type $cx_1^{i_1} x_2^{i_2}... x_n^{i_n}$ where $c$ is constant depending on $i_1,i_1,...,i_n$. Example: $2x_1^{3}-5x_2^{6}$ is a polynomial in $x_1$ and $x_2$.

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    $\begingroup$ So sums of terms that’s are products of integer powers of the variables, with coefficients for each term? $\endgroup$ Commented Jan 30, 2020 at 9:25
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    $\begingroup$ @lukejanicke Yes, exactly. A finite sum, to emphasize. $\endgroup$ Commented Jan 30, 2020 at 9:28
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Hint: This is a question, which you can solve yourself by looking up the definitions in any book on abstract algebra. Start with the polynomial ring $R=\Bbb Z[X]$ in one variable and then define $$ \Bbb Z[X,Y]=R[Y], $$ where $R[Y]$ is the polynomial ring in one variable over the ring $R$ and iterate to obtain the ring of polynomials $\Bbb Z[X_1,\ldots ,X_n]$ in $n$ variables.

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A polynomial with a single indeterminate $x$, denoted $P(x)$, with coefficients in a set $A$ that is $\mathbb Z$, $\mathbb Q$, $\mathbb R$ or $\mathbb C$ is defined as an object of the form $a_0 + a_1 x^1 + a_2 x^2 + ... + a_n x^n$ for some non-negative integer $n$ and for $a_0, ..., a_n \in A$ and $a_n \ne 0$. Denote the set of polynomials with coefficients in $A$ as $A[x]$.

Note: The point of the '$a_n \ne 0$' condition is 1. to say that $0+3x+4x^2$ is the same polynomial as $0+3x+4x^2+0x^3+0x^4$ and 2. To eliminate possibility of infinite sum polynomials such as $0+3x+4x^2+0x^3+0x^4+...$.

For example, a $P(x) \in \ \mathbb Z[x]$ could be $P(x) = 2x^3 + 5$. We also have $P(x) \in \ \mathbb Q[x]$, $ \in \ \mathbb R[x]$, $\in \ \mathbb C[x]$. However for $Q(x) = \pi x$, $Q(x) \in \ \mathbb R[x]$ and $Q(x) \in \ \mathbb C[x]$ but $Q(x) \notin \ \mathbb Q[x]$ and $Q(x) \notin \ \mathbb Z[x]$

A polynomial with two indeterminates $x, y$, denoted $P(x,y)$, with coefficients in $A$ is defined as an object of the form $a_0 + a_1 x^1 + a_2 x^2 + ... + a_n x^n + b_1 y^1 + b_2 y^2 + ... + b_m y^m$ $ + c_{1,1} x^1 y^1 + c_{1,2} x^1 y^2 +$ $ c_{2,1} x^2 y^1 + c_{2,2} x^2 y^2 + ... + c_{p,q} x^p y^q$, for some ... you get the idea, hopefully. More compactly, $P(x,y)$ has the form $d_0 + \sum_{i=0}^p \sum_{j=0}^q d_{i,j}x^iy^j$, with $d_{i,j} \in A$ and for some non-negative integers $p$ and $q$...and then there's the matter of the non-zero term. We must have that $d_{p,q} \ne 0$.

Okay so what about a polynomial with $n$ indeterminates $x_1, ..., x_n$, denoted $P(x_1, ..., x_n)$, with coefficients in $A$? This is defined as an object of the form $\sum_{i_1=0}^{p_1} \ \cdots \ \sum_{i_n=0}^{p_n} d_{i_1, ..., i_n}x^{i_1} \ \cdots \ x^{i_n}$, with $d_{i_1, ..., i_n} \in A$ and for some non-negative integers $p_1, ..., p_n$, where $d_{p_1, ..., p_n} \ne 0$.

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