A functional property of homogeneous polynomials

Let p(x_1,\ldots,x_n) be a homogeneous polynomial of degree k in R[x_1,\ldots,x_n]. Prove that for all \lambda \in R, we have p(\lambda x_1, \ldots, \lambda x_n) = \lambda^k p(x_1,\ldots,x_n).


Recall that a polynomial is called homogeneous of degree k if it can be written in the form p(x_1,\ldots,x_n) = \sum_{i=1}^m r_i \prod_{j=1}^n x_j^{k_{i,j}}, where for each i, we have \sum_{j=1}^n k_{i,j} = k.

This being the case, we have p(\lambda x_1, \ldots, \lambda x_n) = \sum_{i=1}^m r_i \prod_{j=1}^n (\lambda x_j)^{k_{i,j}} = \sum_{i=1}^m r_i \prod \lambda^{k_{i,j}} x_j^{k_{i,j}} = \sum_{i=1}^m r_i \lambda^{\sum k_{i,j}} \prod_{j=i}^n x_j^{k_{i,j}} = \sum_{i=1}^m r_i \lambda^k \prod_{j=1}^n x_j^{k_{i,j}} = \lambda_k \sum_{i=1}^m r_i \prod_{j=1}^n x_j^{k_{i,j}} = \lambda^k p(x_1, \ldots, x_n).

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