Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

Abstract

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space $Ray\left(V\right)$. In particular, these functions induce a partition of $Ray\left(V\right)$ into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.

Keywords:

• Supertropical algebra
• supertropical modules
• bilinear forms
• quadratic forms
• quadratic pairs
• ray spaces
• convex sets
• quasilinear sets
• Cauchy–Schwarz ratio
• Cauchy–Schwarz functions
• QL-stars

2010 Mathematics Subject Classifications:

• Primary: 15A03
• 15A09
• 15A15
• 16Y60
• Secondary: 14T05
• 15A33
• 20M18
• 51M20

Acknowledgments

The authors thank the referee for the helpful suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 $\mathrm{QL}\left(X\right)$ is not necessarily quasilinear.

2 In the following, we omit the factor $e=1_{eR}$, reading all formulas in $eR$.

3 Actually we know this for long, cf. the arguments following (17(17) $\xi =\frac{{\alpha }_{12}}{{\alpha }_{13}}.$(17) ).

4 This also includes information about the zero set of this function.

5 In basic terms, it means that $\mathrm{CS}(W,Z_1)\le \mathrm{CS}(W,Z_2)$ for all $Z_1,Z_2\in [Y_1,Y_2]$ with $[Y_1,Z_1]\subset [Y_1,Z_2]$.

6 In the case $\mathrm{CS}(Y_1,Y_2)\le e$, we also need the information whether the square class $eq\left(Y_1\right)q\left(Y_2\right)\subset \mathcal{G}$ of eR is trivial or not (cf. Theorem 3.3).

7 The reader may argue that our notion of basic type lacks a precise definition. We can remedy this by defining the basic types on $[Y_1,Y_2]$ as all the conditions A, $\mathrm{\partial }A$, B, … appearing in Tables 4.3, 4.4 and Scholium 4.5 below.

8 Although D and $\mathrm{\partial }D$ are the same sentences as E and $\mathrm{\partial }E$ in Table 4.3, we use a different letter ‘D’, since we include in the type the information whether $\mathrm{CS}(Y_1,Y_2)>e$ or $\mathrm{CS}(Y_1,Y_2)\le e$.

9 Taken up to interchanging $Y_1,Y_2$, the type T is listed in Table 4.3.

10 $c_0$ denotes the smallest element $>e$ in $\mathcal{G}$. It exists since $\mathcal{G}$ is discrete.

11 Note that $\alpha {\cong }_{\nu }\text{β}$, $\alpha \in \mathcal{G}\Rightarrow \text{β}\le \alpha$ for any $\alpha ,\text{β}\in R$.

12 The overall assumption that eR is a semifield is not necessary, cf. [2, Definition 7.5].

13 We leave the important problem aside, whether C has a unique minimal set of generators. It would take us too far afield.

14 Recall that ${\mu }_W\left(S\right)={\mu }_W\left(C\right)$ denotes the minimal value of $\mathrm{CS}(W,-)$ on S, and hence on C.

15 Here, it is not necessary to assume that $\mathrm{CS}(Z,W)>0$.