Tropical izatiou in Real AlgebraicGeometry and Extremal Gurbiuatorics

( Or How I learned not to worryand love the tropics ) .

Joint with :

• A . Raymond , M . Singh , R . Thomas

• F . Rincon , R .Sinn

,C. Vinzant

,J. Yu

Se RI . closedall polynomial inequalities valid

ou S'

Xd z XB xd* X Bz o

-

Pure binomial inequalityIn

linear inequality or logsYe -- log Xo

{ Lilli 2 E Bi Yi { (Li -pity,Zo

S → logos)

6nelh.gl#) - right object4 to study

convex Gue-

-

sets with Hadamard Property④ ,

- - Hu) * Hi, - -int -- High - -sxuyu)

x,yes ⇒ x * yes

If S'

has Hadamard propertythen ago) = tropes )-

trop est Liya logics )If S is semi algebraic ⇒tropes) is a rational

polyhedral complex .IAlessandrini )

A = {dy - - , Lu}

Y :L: x t (Xd; . . . xdk)The following preserveHadamard property

(1) Monomial weeps

(2) Convex or comical lull

-

A,B Eo PSD symmetric

wonder

A-* B ko Schur productHun

V -i UVT bae (out)11

lone g PSDmatrices

tropes: ) J .Yu

( ' Yi: ) xoxo- Xiao

Xu Xu Xu Yoo -141, -241020

Also true if mom filledUlmanurialsor pure binomials .

A Pnf - polynomial willsupport n A

wowuegshte on khz o .

Pat -- NTARecord wounds on measures

supported on Rkoat = ( SH 'du, . . .#defbe htt . . Lay

NTA -_ env (yakked)YA :X Hkd's . > Xdk)

drop Ctf ) ?3HnkH9o)+ flirt

10,0) ( 1,21 +fail)f-¥ (2,17

Moha 's Moo - Uh , -Miz( 111 )

Moo - SS da UnifiedMine SXN-ndfemi-fxx.de

then : hoop (UTA) is the one

Of convex punchbars on A .

{*= Bos x xg. Sos t

k -Sos t n tXa .Sos

Etf - will also haveHardwood properly .

trop LEE)thus ( it is all g lattice p b

g a polytope)

trop (Ej) = puncheon orA

hat are midpointconvex

.

-

al I, Lf I Hah

SCH = #EgG

D (G) =Can I

# triangles n C

F)# of

how I→G-

n-

D u,y O- 9312

19"

¥4graph pmfileRazbiov

S - Hadamard propertyU a collection g ↳ invoked

graphtropes) - appear to be a

rational polyhedral6hL

.