Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$xy=0$

$ax +by +cxy +d=0$

$ax +by +cz +dxy +eyz +gxyz=0$

I made myself the examples, sometimes I face these equations and I do not know how to resolve them, all equations whose unknowns have exponent equals one but they can be multiplied together as I have put in the example $xy$, $yxzt$.... I want to know the name so I can find info and understand them because I google equations and many different come, mostly linear but I do not see these ones. I did not see the tag "equations" so I tagged differential-equations but I do not think It is that.

share|improve this question
2  
I suppose the term for such equations would be "systems of multilinear equations". –  Raskolnikov Jan 17 '12 at 14:03

1 Answer 1

up vote 0 down vote accepted

In terms of nomenclature, equations of the form $$ c_1 x_1 + c_2 x_2 + \ldots $$

where $c$ is a constant are known as linear equations, equations: $$ c_1 x_1 x_2 + c_2 x_2 x_3 + \ldots $$

are bilinear equations. Though not as common, equations of the form: $$ c_1 x_1 x_2 x_3 + c_2 x_2 x_3 x_4 + \ldots $$ are referred to as trilinear equations. For higher orders @Raskolnikov suggested "systems of multilinear equations". Also, see the comment by @Lieven below who calls them "multilinear equations in homogeneous space".

Solving systems of linear equations is one of the goals of Linear Algebra. Less is known for bilinear equations, but such systems can often be "solved" in the sense of finding a near optimal solution (ie. as a constraint problem).

share|improve this answer
    
The examples he posted aren't multilinear, because in that case, all terms should be a product of the same number of variables. All examples the OP posted are multivariate polynomials, although multivariate polynomials can of course also contain higher exponents of the same variable in the same term. They could be considered multilinear equations in homogeneous space though. –  Lieven Jan 17 '12 at 14:42
    
Thank you, is there any way to solve them with exact solutions? –  user23090 Jan 21 '12 at 3:45
    
@user23090 I suspect each case has a different answer, but in general the answer is no. If you were looking to solve your first two equations (the bilinear ones), I would ask that as a separate question. –  Hooked Jan 22 '12 at 5:46

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.