Polytope approach for solving ultradiscrete ordinary difference equations
Yoichi Nakata1
1Isotope Science Center, the University of Tokyo
Abstract

Difference equations
Laurent polynomials
X

aI x IS ⇢ Zd<latexit sha1_base64="2Rd/rzoV5pOZsjx8TG1J4S1DPHg=">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</latexit>

I2S

Ultradiscrete systems|S| < 1max(AI + hI, Xi)u ad sc e za on<latexit sha1_base64="cZZOZStbyrkL1zF04as8KRm5CtA=">AAACb3ichVHLSsNAFD2Nr1pfVRcKggSLopsy1YJWXIhuXGprVbAiSZzWwTQJybRQqz/gB+jChQ8QET/DjT/gop8grkTBjQtv04iIqHeYzJkz99w5k6s7pvAkY7WQ0tTc0toWbo90dHZ190R7+9Y8u+QaPGvYpu1u6JrHTWHxrBTS5BuOy7WibvJ1fW+xfr5e5q4nbGtVVhy+VdQKlsgLQ5NE5Q4yB+qcmhNWXla2ozEWZ36oP0EiADEEsWxHr5HDDmwYKKEIDguSsAkNHo1NJMDgELeFKnEuIeGfcxwiQtoSZXHK0Ijdo2+BdpsBa9G+XtPz1QbdYtJ0SalilD2wG/bC7tkte2Tvv9aq+jXqXiq06g0td7Z7jgYzb/+qirRK7H6p/vQskceM71WQd8dn6q8wGvry/slLZjY9Wh1jl+yJ/F+wGrujF1jlV+NqhadPEfEbkEolp5JJ9Sf4bMDaZDxBeCUZm18IWhHGEEYwTv97GvNYwjKydK+DY5zhPPSsDCjDitpIVUKBph/fQpn4AHv+jps=</latexit>

I2S<latexit sha1_base64="m6Q0a5zRw7uvOYmwpchBuiAqSZQ=">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</latexit>

1 1 co espondence

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I = (I1 , . . . , Id )Polytope representations
Convex hull of I (Newton polygons of polynomials)

x = (x1 , . . . , xd )<latexit sha1_base64="Wbam/8yaSGCL5BidROb0fJxdyrU=">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</latexit><latexit sha1_base64="Y6VZ4gx7B96nFapEl/z1MNI08ZE=">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</latexit><latexit sha1_base64="EtrcrWqJie+F+wk5Yn3+r4OvnII=">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</latexit>

f+g (sum of two polynomials)

max(F, G) = max (A + hI Xi A + hI Xi)

fg (product of two polynomials)

F+G =Convex hull of union of two polytopes

2S

max (A + A + hI + I Xi)

Minkowski sum of two polytopes

2S

Laurent phenomenon for evolution equations

Laurent phenomenon for ultradiscrete evolution
equationsfn fn N = (fn 1 , . . . , fn N +1 )<latexit sha1_base64="LzIsI9bsRcwmMB5Lk3LHyIa0y8g=">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</latexit>

RHS Fn N as –max free form can be obtained by
the inverse calculation of obtaining the Minkowski
Fn + Fn N = (Fn 1 , . . . , Fn N +1 ) sum.
RHS Fn N should be expressed as the form:
max A + I X -max free!
<latexit sha1_base64="DUWfQmCZfUra4VqjVa6AXMeRunQ=">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</latexit>

All fn s are Laurent polynomials of the initial values[2](RHS is divisible by fn N ).
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2S

Polynomial f is divisible by g.

Some mechanism to realize
F – G = max A + I X
2S
where F= max A + I X , G= max A + I X .
2S
2SWhat mechanism?

Existence of a polytope H satisfying F + H = G,
where F and G are polytopes and + means Minkowskisum.
We find an algorithm to calculate the Minkowski sum
and its inverse by focusing on the edges of polytopes.

Mot vat on

Resu tAlgorithm to obtain Minkowski sum for polygons in two
dimensional space.

F5 = 2 max(6B 0 4A 3A + 2B)n he case wh ch he deg ee o eedom n he n a va ue p ob em o heu ad sc e e sys em s wo he so u on s exp essed as wo n a va ues
wh ch co esponds o a po y ope Fo po ygons n wo d mens ona space
he e s a s mp e a go hm o ca cu a e M nkowsk sum and s nve se
A go hm
1
Take he un on o edge vec o se s o wo po ygons as mu se s
and so he edge vec o s by he a gumen
2
he e a e wo edge vec o s sa s y ng u = kv (k > 0)
ep ace hem w h u + v
Examp ePolygonEdge vector set= 2 max(4B 0 3A)

4A

3B

1)B 0 (n2)A)

2 max(2B 0 A)

4A

3B

Gene a y we have

Fn = 2 max((n

(n

1)A

(n

2)B (n2)

Examp e B n eg ab e squa e o de g ow h

Fn + Fn 4 = max(Fn 1 + Fn 3 2Fn 2 ) (nF0 = F1 = 0 F2 = A F3 = B

4)The po ygons co espond ng o he so u on Fn s decomposed n o a quad ang ewhose sca e g ows squa e o de and he po ygon w h pe od 8 3 4
Examp e C non n eg ab e bu w h Lau en p ope y exponen a g ow h

04

Fn + Fn 2 = 3 max(Fn 2 0)(0,-4), (3,0), (-3,4)
00

30

+

Fn = 2 max(bn 1 B 0 bn 2 A)
(0,-2), (1,0), (-1,2)

10

an 2 B(n2)

an = 3an 1

an 2

a 1 = 0 a0 =bn = 3bn 1

bn 2

b1 = 3 b0 = 01
espec ve y

Laurent phenomenon for ultradiscrete equations with higher degree of freedom
3200

an 1 A=06

2)

whe e an and bn a e he so u ons o

02
00

(n

F0 = A F1 = B(0,-6), (4,0), (-1,2), (-3,4)

Ou a go hm can be ex ended o he ca cu a on o h ghe d mens ona
po y opes We can ca cu a e n a va ue p ob em w h h ghe deg ee o eedom

40

Examp emax(4B 0 3A) + max(2B 0 A) = max(6B 0 4A 3A + 2B)

Fn + Fn 6 = max(Fn 1 + Fn 5 Fn 2 + Fn 4 2Fn 3 ) (n6)

The nve se o M nkowsk sum s ca cu a ed by app y ng h s me hod nve se y

F0 = F1 = 0 F2 = A F3 = B F4 = C F5 = D

Laurent phenomenon for two degree of freedom ultradiscrete system.

Problem
No explicit formulae! Number of the vertices of the polytope grows as nincreases because the solution Fn is decomposed into a square order growthterm and the periodic term whose period depends on the initial value.
Finding equations which our method fits is the future work.

By v ue o app y ng h s me hod o seve a u ad sc e e equa ons we
ob a ned he exp c o mu a o he so u on o he n a va ue p ob em
Examp e A L nea zab e L nea g ow h

Re e ences

Fn + Fn 2 = 2 max(Fn 1 0) (n2)

F0 = A F1 = B
F2 = 2 max(B 0) A
F3 = 2 max(2B 0 A) 2A

B

F4 = 2 max(4B 0 2A 2A + B)
= 2 max(3B 0 2A)

3A

2 max(B 0)

2B

3A

2B

1 Y Naka a The o u on o he n a va ue p ob em o he u ad e e Somo 4 and 5
equa on a X v 1701 04262
2 S Fom n and A Ze ev n ky The au en phenomenon Adv n App Ma h 28 119–144 2002
3 A P Fo dy and A Hone D e e n eg ab e Sy em and Po on A geb a F om C u e
Map Commun Ma h Phy 325 527–584 2014
4 A P Fo dy and A Hone Symp e
Map om C u e A geb a S GMA 2011
Vo ume 7 091