Is x²-y²+3xy-x⁴+5y⁴+z⁴-2t⁴+3xyzt-6x²tz+5x²y² a 4-Linear Form?

[Sangoku]

Is this a 4-linear form ??

rather a silly question, but i am a bit confused , could we consider the expression

$$x^{2}-y^{2}+3xy-x^{4}+5y^{4}+ z^{4}-2t^{4}+3xyzt-6x^{2}tz+ 5x^{2}y^{2}$$

or even in general $$A_{i,j}x^{i}x^{j}-Q_{i,j,k,l}x^{i}x^{j}x^{k}x^^{l}$$

could be considered a 4-linear form??.. i believe this is linear on every argument x^{i} but there seems to be 'mixed up' a 2-from plus a 4-form so i,m not sure what exactly is the proposed object.

 

[mathman]

The usual label for such expressions is 4th degree polynomial in 4 variables.

 

[tacman]

No, neither of these expressions can be considered a 4-linear form. A 4-linear form is a function that takes in four vectors and outputs a scalar value, and it must be linear in each of its arguments. In the first expression, there are terms that are not linear in their arguments (such as x^4 and y^4), and in the second expression, there is a mix of 2-form and 4-form terms, which does not fit the definition of a 4-linear form.