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Abstract

When we are asked to write the polynomial p(x)=3x2+x4+7−xp(x) = 3x^2 + x^4 + 7 - xp(x)=3x2+x4+7−x in standard form, we know our answer is x4+3x2−x+7x^4 + 3x^2 - x + 7x4+3x2−x+7, because the only way to order powers of xxx totally while being consistent with multiplication is the natural one: degree-wise!

Complexity is introduced when we are asked to write q(x)=2x3yz2−6x4z+x2y3zq(x) = 2x^3yz^2 - 6x^4z + x^2y^3zq(x)=2x3yz2−6x4z+x2y3z in standard form, and we have to define what standard form means in this setting.

In this talk, we will introduce the concept of Monomial Orders through examples, discuss the basics of their matrix representations, present the three classical monomial orders (lexicographic, degree lexicographic, and reverse lexicographic), explore a curious property they possess (hint: it is in the title), and present results found in joint work with undergraduates in 2016 and 2023 at Amherst College and Colgate University, respectively, seeking to characterize other monomial orders possessing this 'reconstructible' property.

The only background needed for (most of) this talk is basic linear algebra!