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Define a partitioned homogeneous polynomial of degree $d$ to be a polynomial in $$\mathbb Z[x_{11},\dots,x_{1n},\dots,x_{d1},\dots,x_{dn}]$$ with monomials from entries in (polynomials that are $d$-linear forms of a particular structure) vector $$(x_{11},\dots,x_{1n})\otimes\dots\otimes(x_{d1},\dots,x_{dn}).$$

  1. Supposing if $f,g$ are two such polynomials with coefficient vectors linearly independent then when does it hold that they are algebraically independent?

  2. Will it work if $\mathbb Z$ is replaced by another commutative ring?

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    $\begingroup$ I'm not sure I understand your notation - do you mean that the polynomial is a linear combination of monomials of the form $x_{1a_1}x_{2a_2}\dots x_{da_d}$ with $1\leq a_k\leq n$? $\endgroup$
    – Wojowu
    Commented Jan 17, 2019 at 20:37
  • $\begingroup$ @Wojowu Yes that is correct. $\endgroup$
    – Turbo
    Commented Jan 17, 2019 at 20:51
  • $\begingroup$ In other words, it is a $d$-linear form on $\mathbb{Z}^n$, right? $\endgroup$
    – Laurent Moret-Bailly
    Commented Jan 18, 2019 at 8:23
  • $\begingroup$ @LaurentMoret-Bailly Yes that is correct. $\endgroup$
    – Turbo
    Commented Jan 18, 2019 at 9:02
  • $\begingroup$ Well, any $d$-linear form is a linear combination of forms of the shape given in Wojowu's comment. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jan 19, 2019 at 19:03

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