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This may seem trivial, but I need to find a name for the following function for a publication.

Its general $N$-dimensional form is $$y = \sum_{i=0}^{2^N-1} a_i \prod_{j=0}^{N-1} x_{j+1}^{\left \lfloor i/2^j \right \rfloor \mod 2}$$ where $\lfloor x \rfloor$ indicates the floor (round down) operator.

When $N=1$ the function is a straight line $y = a_0 + a_1 x_1$ and when $N=2$ it's a parabolic hyperboloid $y = a_0 + a_1 x_1 + a_2 x_2 + a_3 x_1 x_2$. In the general case it maintains at least one of the properties of the parabolic hyperboloid, i.e. by fixing the value of $N-1$ variables, the function becomes a straight line in the $N$-th variable: $y = A_0 + A_1 x_k$.

I'm not sure I can call it an $N$-dimensional parabolic hyperboloid, because the PH is a 2nd degree function (it can be written alternatively as $y = b_1 x_1^2 - b_2 x_2^2$, with $b_1 b_2 > 0$ plus some translations for the two variables), while the general $N$-dimensional form should be of $N$-th degree.

Is there a proper name I can give to this function? If necessary, I'm particularly interested in the $N=4$ case.

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This is a good question. I am not sure it has an accepted name, but I would name it an "elementary linear polynomial function in $\ n \ $ variables" because linear combinations of elementary symmetric polynomials are an important special case where the coefficients of each homogeneous part are equal. The fundamental feature is that you allow the function to be at most linear in each variable separately which is a restriction of the general polynomial function in several variables with no such restriction on the degree of each variable separately.

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