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=== Equivalent definition ===
Polynomials in ''n'' variables can also be defined as functions from '''N'''<sup>''n''</sup> into ''R'' (where it is important that '''N''' should be taken to contain 0) which are zero everywhere except for a finite number of points, with the addition and ''R''-multiplication defined in the canonical way, and multiplication defined by the [[convolution]]
:<math>P * Q : k\mapsto\sum_{i+j=k} P_i\,Q_j ~,</math>
where ''i,j,k''∈'''N'''<sup>''n''</sup> are the [[multi-index|(multi-)indices]] corresponding to respective powers of the indeterminates (and <math>P_i,Q_j</math> are the associated ''coefficients'' of the respective polynomial).
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