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The notation in Fulton's Algebraic Curves 1.7(a) is confusing me to the point where I cannot even make examples in order to begin:

Let $k$ be a field, $F\in k[X_1,\ldots,X_n]$, $a_1,\ldots,a_n\in k$. Show that

$$F=\sum\lambda_{(i)}(X_1-a_1)^{i_1}\ldots(X_n-a_n)^{i_n}\hspace{10px},\hspace{10px}\lambda_{(i)}\in k.$$

Can someone give an example polynomial written into this form so I can make sense of the odd indices?

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1 Answer

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$\lambda_{(i)}$ here refers to a constant which may change depending on the individual powers of each $X_i - a_i$ appearing in the term. That is, $(i) = (i_1,i_2,\dots, i_n),$ so $\lambda_{(i)}$ depends on each $i_j.$

As an example, let $n = 3,$ $a_1 = -1,$ $a_2 = 0,$ and $a_3 = 1,$ and let $k = \Bbb C.$ Such a polynomial is $$ F(X_1,X_2,X_3) = (X_1 + 1)(X_2)(X_3 - 1) + 2(X_1 + 1)(X_3 - 1) - 7(X_2)^3(X_3 - 1)^{12}. $$ Here $\lambda_{(1,1,1)} = 1,$ $\lambda_{(1,0,1)} = 2,$ and $\lambda_{(0,3,12)} = -7.$

You're writing the polynomial in a sort of Taylor expansion at the point $(a_1,\dots, a_n)\in k^n,$ and the $\lambda_{(i)}$ are the coefficients in this expansion as $(i) = (i_1,\dots, i_n)$ ranges through all the elements of $\Bbb N_0^n.$ Writing a given polynomial $p$ in this form in the situation $k = \Bbb R,$ $n = 1$ recovers the usual Taylor expansion of $p$ about the point $X = a.$

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