# Finding the X largest positive and negative variables in any given algebraic expression

Is there a method where given an algebraic expression one can determine which variables within the expression contribute most to the expression's final result, either positively or negatively? There can be multiple instances of each variable within the expression, which must be treated as a single variable (i.e. their effects average).

For example, take the expression: x = a3 + b2 - c

In this case, assuming a, b and c have the same real, >1 possible value range distributions, a has the largest positive effect of the value of x and c has the largest negative value effect.

Extra question: if the above exists, is there further functionality to restrict the value range of certain variables. e.g. a can only be in the range {0-1}, b in the set {0, 1, ... , 99, 100}, etc.

Apologies if this has already been asked. My range of search terms may be the limiting factor. Will delete the question if duplicate found. Appreciate any help offered. Thanks.

• I think your assumption that " assuming a, b and c have the same real, positive possible value range distributions, a has the largest positive effect" is not correct. If, for example, the range is limited to (0..1) then b would on average have a larger positive effect than a. – Penguino Oct 31 '18 at 23:13
• Sorry, I intended 'the same' to indicate a/b/c have the same distributions. I.e. if a is (0...1), b is (0...1) – Jordan MacLachlan Oct 31 '18 at 23:22
• That was precisely my point. If a and b are both chosen uniformly from (0..1) then $a^3$ will on average be smaller than $b^2$, not larger. – Penguino Oct 31 '18 at 23:43
• Apologies - bad day. Good call. Edited question. – Jordan MacLachlan Oct 31 '18 at 23:45

Let's formalize: take some functions $$f_i:\mathbb{R}\to\mathbb{R}$$, with $$i\in\{1,\dots,n\}$$ and consider the function $$x:\mathbb{R}^n\to\mathbb{R}:(a_1,\dots,a_n)\mapsto f(a_1)+\dots+f(a_n)$$. If given ranges for each $$a_i$$, we can figure out the range of $$f_i$$ (more precisely, the image of the function restriction of $$f_i$$ to its respective range). This of course assumes you have enough information about every $$f_i$$ to figure that out.
Edit: this bit only applies if you ignore that the functions must be algebraic expressions. I'm guessing this isn't important, since this answer is currently accepted. The answers to this post (OP requests currently unknown natural constants) provide us with some inspiration for functions for which the maximum or minimum is (currently) not known. For example: denote the $$n$$-th prime number with $$p_n$$, and let $$f:\mathbb{N}\setminus\{0\}\to\mathbb{R}: n\mapsto p_{n+1}-p_n$$ be the sequence of differences of two successive prime numbers. We know that $$\mathrm{max}\{f(n)\:|\: n\in\mathbb{N}\} \leqslant 246$$ (source), but we don't know if $$246$$ is really the maximum. Hence we don't know what would be the largest positive effect that $$f$$ can produce.
• I didn't see that you specifically asked for algebraic expressions. In the example I gave I don't think (read: I guess it can't, but I'm not sure) $f$ can be written as an algebraic expression. Maybe some kind of method exists for algebraic expressions. – Aldoggen Oct 31 '18 at 23:40