# What is meant by the notation “a polynomial $p(x_1, ..., x_n)$”? [duplicate]

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.”

• 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 $\mathbf Z[x, y,z]$. Commented Jan 30, 2020 at 9:15
• @DietrichBurde that looks similar to what my searches turned up. Was hoping for something not in abstract algebra terms. Commented Jan 30, 2020 at 9:20
• 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. Commented Jan 30, 2020 at 9:29

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 $$x_2$$.

• So sums of terms that’s are products of integer powers of the variables, with coefficients for each term? Commented Jan 30, 2020 at 9:25
• @lukejanicke Yes, exactly. A finite sum, to emphasize. Commented Jan 30, 2020 at 9:28

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.

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 $$d_{p_1, ..., p_n} \ne 0$$.