Necessary conditions for the invariant measure of a random walk to be a sum of geometric terms

Memorandum 2006 (April 2013). ISSN 1874−4850. Available from: http://www.math.utwente.nl/publications Department of Applied Mathematics, University of Twente, Enschede, The Netherlands

◆❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢

❛ r❛♥❞♦♠ ✇❛❧❦ t♦ ❜❡ ❛ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s

❨❛♥t✐♥❣ ❈❤❡♥✶_{✱ ❘✐❝❤❛r❞ ❏✳ ❇♦✉❝❤❡r✐❡}✶ _{❛♥❞ ❏❛s♣❡r ●♦s❡❧✐♥❣}✶✱✷ ✶_{❙t♦❝❤❛st✐❝ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❯♥✐✈❡rs✐t② ♦❢ ❚✇❡♥t❡✱❚❤❡} ◆❡t❤❡r❧❛♥❞s ✷_{❉❡♣❛rt♠❡♥t ♦❢ ■♥t❡❧❧✐❣❡♥t ❙②st❡♠s✱ ❉❡❧❢t ❯♥✐✈❡rs✐t② ♦❢} ❚❡❝❤♥♦❧♦❣②✱❚❤❡ ◆❡t❤❡r❧❛♥❞s ④②✳❝❤❡♥✱r✳❥✳❜♦✉❝❤❡r✐❡✱❥✳❣♦s❡❧✐♥❣⑥❅✉t✇❡♥t❡✳♥❧ ❆♣r✐❧ ✶✶✱ ✷✵✶✸ ❆❜str❛❝t ❲❡ ❝♦♥s✐❞❡r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ ❤♦♠♦❣❡♥❡♦✉s r❛♥❞♦♠ ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❝♦♥s✐❞❡r ♠❡❛s✉r❡s t❤❛t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ❲❡ ♣r❡s❡♥t ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ ❛ r❛♥❞♦♠ ✇❛❧❦ t♦ ❜❡ ❛ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ❲❡ ❞❡♠♦♥str❛t❡ t❤❛t ❡❛❝❤ ❣❡♦♠❡tr✐❝ t❡r♠ ♠✉st ✐♥❞✐✈✐❞✉❛❧❧② s❛t✐s❢② t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✳ ❲❡ s❤♦✇ t❤❛t t❤❡ ❣❡♦♠❡tr✐❝ t❡r♠s ✐♥ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♠✉st ❤❛✈❡ ❛ ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ str✉❝t✉r❡✳ ❲❡ ❢✉rt❤❡r s❤♦✇ t❤❛t t❤❡ r❛♥❞♦♠ ✇❛❧❦ ❝❛♥♥♦t ❤❛✈❡ tr❛♥s✐t✐♦♥s t♦ t❤❡ ◆♦rt❤✱ ◆♦rt❤❡❛st ♦r ❊❛st✳ ❋✐♥❛❧❧②✱ ✇❡ s❤♦✇ t❤❛t ❢♦r ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s t♦ ❜❡ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❛t ❧❡❛st ♦♥❡ ❝♦❡✣❝✐❡♥t ♠✉st ❜❡ ♥❡❣❛t✐✈❡✳ ❚❤✐s ♣❛♣❡r ❡①t❡♥❞s ♦✉r ♣r❡✈✐♦✉s ✇♦r❦ ❢♦r t❤❡ ❝❛s❡ ♦❢ ✜♥✐t❡❧② ♠❛♥② t❡r♠s t♦ t❤❛t ♦❢ ❝♦✉♥t❛❜❧② ♠❛♥② t❡r♠s✳ ❑❡②✇♦r❞s✿ r❛♥❞♦♠ ✇❛❧❦❀ q✉❛rt❡r✲♣❧❛♥❡❀ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡❀ ❣❡♦♠❡tr✐❝ t❡r♠❀ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡❀ ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞❀ ❝♦♠♣❡♥s❛t✐♦♥ ♠❡t❤♦❞ ✷✵✶✵ ▼❛t❤❡♠❛t✐❝s ❙✉❜❥❡❝t ❈❧❛ss✐✜❝❛t✐♦♥✿ ✻✵❏✶✵❀ ✻✵●✺✵

✶ ■♥tr♦❞✉❝t✐♦♥

❘❛♥❞♦♠ ✇❛❧❦s ❢♦r ✇❤✐❝❤ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✐s ❛ ❣❡♦♠❡tr✐❝ ♣r♦❞✉❝t ❢♦r♠ ❛r❡ ♦❢t❡♥ ✉s❡❞ t♦ ♠♦❞❡❧ ♣r❛❝t✐❝❛❧ s②st❡♠s✳ ❚❤❡ ❜❡♥❡✜t ♦❢ ✉s✐♥❣ s✉❝❤ ♠♦❞❡❧s P♦st❛❧ ❛❞❞r❡ss✿ ❙t♦❝❤❛st✐❝ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣✱ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❯♥✐✈❡rs✐t② ♦❢ ❚✇❡♥t❡✱ P✳❖✳ ❇♦① ✷✶✼✱ ✼✺✵✵ ❆❊ ❊♥✲ s❝❤❡❞❡✱ ❚❤❡ ◆❡t❤❡r❧❛♥❞s✳ ✶

✐s t❤❛t t❤❡✐r ♣❡r❢♦r♠❛♥❝❡ ❝❛♥ ❜❡ ❡❛s✐❧② ❛♥❛❧②③❡❞ ✇✐t❤ tr❛❝t❛❜❧❡ ❝❧♦s❡❞✲❢♦r♠ ❡①♣r❡ss✐♦♥s✳ ❍♦✇❡✈❡r✱ t❤❡ ❝❧❛ss ♦❢ r❛♥❞♦♠ ✇❛❧❦s ✇❤✐❝❤ ❤❛✈❡ ❛ ♣r♦❞✉❝t ❢♦r♠ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ✐s r❛t❤❡r ❧✐♠✐t❡❞✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♦❢ ✐♥t❡r❡st t♦ ✜♥❞ ❧❛r❣❡r ❝❧❛ss❡s ♦❢ tr❛❝t❛❜❧❡ ♠❡❛s✉r❡s t❤❛t ❝❛♥ ❜❡ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❢♦r r❛♥❞♦♠ ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡✳ ■♥ ♣r❡✈✐♦✉s ✇♦r❦ ✇❡ ❝♦♥s✐❞❡r❡❞ t❤❡ ♠❡❛s✉r❡s t❤❛t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✜♥✐t❡❧② ♠❛♥② ❣❡♦♠❡tr✐❝ t❡r♠s ❬✻❪✳ ■♥ t❤❡ ❝✉rr❡♥t ✇♦r❦ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ ♦❢ ❝♦✉♥t❛❜❧② ♠❛♥② t❡r♠s✳ ❲❡ st✉❞② r❛♥❞♦♠ ✇❛❧❦s ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡ t❤❛t ❛r❡ ❤♦♠♦❣❡♥❡♦✉s ✐♥ t❤❡ s❡♥s❡ t❤❛t tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❛r❡ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✳ ❖✉r ✐♥t❡r❡st ✐s ✐♥ ✜♥✐t❡ ♠❡❛s✉r❡s m(i, j) t❤❛t ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s m(i, j) = X (ρ,σ)∈Γ α(ρ, σ)ρi_σj_{, ✇✐t❤ |Γ| = ∞, ✭✶✮} ✐✳❡✳✱ m(i, j) ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ❛s ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ❈♦♥tr❛r② t♦ ♠✉❝❤ ♦t❤❡r ✇♦r❦✱ ❢♦r ✐♥st❛♥❝❡ ❬✼✱ ✾✱ ✶✶✱ ✶✷❪✱ ♦✉r ✐♥t❡r❡st ✐s ♥♦t ✐♥ ✜♥❞✐♥❣ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r s♣❡❝✐✜❝ r❛♥❞♦♠ ✇❛❧❦s✳ ■♥st❡❛❞ ♦✉r ✐♥t❡r❡st ✐s ✐♥ ❝❤❛r❛❝t❡r✐③✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ♣r♦♣❡rt✐❡s ♦❢ r❛♥❞♦♠ ✇❛❧❦s✱ s❡ts Γ ❛♥❞ ❝♦❡✣❝✐❡♥ts α t❤❛t ❛❧❧♦✇ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t♦ ❜❡ ❡①♣r❡ss❡❞ ✐♥ ❢♦r♠ ✭✶✮✳ ❚❤❡ ❝♦♥tr✐❜✉t✐♦♥s ♦❢ t❤❡ ❝✉rr❡♥t ✇♦r❦ ❛r❡ ❛ s❡t ♦❢ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s✿ ✶✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ Γ ♠✉st ✐♥❞✐✈✐❞✉❛❧❧② s❛t✐s❢② t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✳ ✷✳ ❈♦♥s❡❝✉t✐✈❡ ❡❧❡♠❡♥ts ♦❢ Γ ♠✉st ❤❛✈❡ ❛ ❝♦♠♠♦♥ ❝♦♦r❞✐♥❛t❡✱ ✐✳❡✳✱ Γ ♠✉st ❤❛✈❡ ❛ str✉❝t✉r❡ ✇❤✐❝❤ ✇❡ r❡❢❡r t♦ ❛s ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞✳ ✸✳ ■♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✱ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ❤❛s ♥♦ tr❛♥s✐t✐♦♥s t♦ t❤❡ ◆♦rt❤✱ ◆♦rt❤❡❛st ♦r ❊❛st✳ ✹✳ ❆t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts α(ρ, σ) ✐♥ ✭✶✮ ♠✉st ❜❡ ♥❡❣❛t✐✈❡✳ ❚❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡ ✐♥❞✉❝❡ ❛♥ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡✳ ❆ ❝❧♦s❡❧② r❡❧❛t❡❞ ❝✉r✈❡ ❛r✐s❡s ❛s t❤❡ ❦❡r♥❡❧ ♦❢ t❤❡ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠s st✉❞✐❡❞ ✐♥ ❬✼✱ ✾❪ ❛♥❞ r❡❧❛t❡❞ ✇♦r❦✳ ❙♦♠❡ ♦❢ ✐ts ❜❛s✐❝ ♣r♦♣❡rt✐❡s ✇❡r❡ ❞❡r✐✈❡❞ ✐♥ ❬✾❪✳ ❆♥ ✐♠♣♦rt❛♥t ♣❛rt ♦❢ t❤❡ ❝✉rr❡♥t ✇♦r❦ ❝♦♥s✐sts ♦❢ st✉❞②✐♥❣ t❤✐s ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ ✐♥ ♠♦r❡ ❞❡t❛✐❧✳ ❙❡❝t✐♦♥ ✸ ♣r❡s❡♥ts ♥❡✇ r❡s✉❧ts ♦♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤✐s ❝✉r✈❡✳ ❆❞❛♥ ❡t ❛❧✳ ❬✶✱ ✷✱ ✸✱ ✹❪ ✉s❡ ❛ ❝♦♠♣❡♥s❛t✐♦♥ ❛♣♣r♦❛❝❤ t♦ ❝♦♥str✉❝t ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t❤❛t ✐s ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤✐s ✐s ❞♦♥❡ ❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ t❤❛t ❤❛✈❡ ♥♦ tr❛♥s✐t✐♦♥s t♦ t❤❡ ◆♦rt❤✱ ◆♦rt❤❡❛st ❛♥❞ ❊❛st✳ ■♥ ❛❞❞✐t✐♦♥ t❤❡ ❣❡♦♠❡tr✐❝ t❡r♠s ✐♥ t❤❡ ♠❡❛s✉r❡ ✇✐❧❧ ✐♥❞✐✈✐❞✉❛❧❧② s❛t✐s❢② t❤❡ ✐♥t❡r✐♦r ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ❛♥❞ ♥❡❣❛t✐✈❡ ✇❡✐❣❤ts ✇✐❧❧ ♦❝❝✉r ✐♥ t❤❡ s✉♠✳ ■♥ t❤❛t s❡♥s❡✱ t❤❡ r❡s✉❧ts ❢r♦♠ ❬✸❪ ❝♦♠♣❧❡♠❡♥t t❤❡ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s t❤❛t ❛r❡ ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r✳ ✷

■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♣♦✐♥t ♦✉t t❤❛t t❤❡ ♠❡❛s✉r❡s t❤❛t ❛r❡ ♦❜t❛✐♥❡❞ ✐♥ ❬✸❪ ❝❛♥ ❝♦♥s✐st ♦❢ ♠✉❧t✐♣❧❡ ♣❛✐r✇✐s❡ ❝♦✉♣❧❡❞ s❡ts ❛♥❞ t❤❛t t❤❡s❡ ♠❡❛s✉r❡s✱ t❤❡r❡❢♦r❡✱ ❞♦ ♥♦t s❛t✐s❢② ♦✉r ❝♦♥❞✐t✐♦♥ 2 ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✱ ✇❤✐❝❤ st❛t❡s t❤❛t ♦♥❧② ❛ s✐♥❣❧❡ ♣❛✐r✇✐s❡ ❝♦✉♣❧❡❞ s❡t ❝❛♥ ♦❝❝✉r✳ ❚❤❡r❡ ✐s ♥♦ ❝♦♥tr❛❞✐❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡s❡ r❡s✉❧ts✱ s✐♥❝❡ t❤❡ ♠❡❛s✉r❡ t❤❛t ✐s ❝♦♥str✉❝t❡❞ ✐♥ ❬✸❪ ✐s ♥♦t ✜♥✐t❡✳ ■♥ ❢❛❝t✱ t❤❡ ♦♥❧② ✜♥✐t❡ ♠❡❛s✉r❡s t❤❛t ❤❛✈❡ ❜❡❡♥ ❝♦♥str✉❝t❡❞ ✉s✐♥❣ t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ❛♣♣r♦❛❝❤ ❝♦♥s✐st ♦❢ ❛ s✐♥❣❧❡ ♣❛✐r✇✐s❡ ❝♦✉♣❧❡❞ s❡t✳ ◆♦t❡✱ ✐♥ ❛❞❞✐t✐♦♥ t❤❛t t❤❡ r❛♥❞♦♠ ✇❛❧❦s st✉❞✐❡❞ ✐♥ ❬✶✱ ✷✱ ✸✱ ✹❪ ❞♦ ♥♦t s❛t✐s❢② ♦✉r ❤♦♠♦❣❡♥❡✐t② ❝♦♥❞✐t✐♦♥s✳ ❲❡ ❞✐s❝✉ss t❤❡s❡ ❝♦♥❞✐t✐♦♥s ❛♥❞ t❤❡ r❡❧❛t✐♦♥ t♦ t❤❡ ❝♦♠♣❡♥s❛t✐♦♥ ❛♣♣r♦❛❝❤ ✐♥ ♠♦r❡ ❞❡t❛✐❧ ✐♥ ❙❡❝t✐♦♥ ✺✳ ❋♦r t❤❡ r❡✢❡❝t❡❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥ ✐♥ ❛ ✇❡❞❣❡✱ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t❤❛t ✐s ❛ s✉♠ ♦❢ ✜♥✐t❡❧② ♠❛♥② ❡①♣♦♥❡♥t✐❛❧s ✇❛s st✉❞✐❡❞ ❜② ❉✐❡❦❡r ❡t ❛❧✳ ❬✽❪✳ ❚❤❡② s❤♦✇ t❤❛t ❢♦r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t♦ ❜❡ ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ✜♥✐t❡❧② ♠❛♥② ❡①♣♦♥❡♥t✐❛❧ ♠❡❛s✉r❡s✱ t❤❡r❡ ♠✉st ❜❡ ❛♥ ♦❞❞ ♥✉♠❜❡r ♦❢ t❡r♠s t❤❛t ❤❛✈❡ ❛ ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ str✉❝t✉r❡✳ ❚❤❡ ♠❡t❤♦❞ ❢♦r t❤❡ ❝♦♥t✐♥✉♦✉s st❛t❡ s♣❛❝❡ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✱ ❤♦✇❡✈❡r✱ ❝❛♥♥♦t ❜❡ ✉s❡❞ ❢♦r t❤❡ ❞✐s❝r❡t❡ st❛t❡ s♣❛❝❡ r❛♥❞♦♠ ✇❛❧❦✳ ■♥ ❬✻❪ ✇❡ ✐♥✈❡st✐❣❛t❡❞ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t❤❛t ✐s ❛ s✉♠ ♦❢ ✜♥✐t❡❧② ♠❛♥② ❣❡♦♠❡tr✐❝s ❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡✳ ❚❤❡ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❞❡r✐✈❡❞ ✐♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r ❢♦r♠ t❤❡ ♥❛t✉r❛❧ ❡①t❡♥s✐♦♥ ♦❢ ♦✉r r❡s✉❧ts ❢r♦♠ ❬✻❪ t♦ t❤❡ ❝❛s❡ ♦❢ ❝♦✉♥t❛❜❧② ♠❛♥② t❡r♠s✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣❛♣❡r ✐s str✉❝t✉r❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷ ✇❡ ♣r❡s❡♥t t❤❡ ♠♦❞❡❧✳ ❆♥ ✐♠♣♦rt❛♥t ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ ❤❛s ❜❡❡♥ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❙❡❝t✐♦♥ ✸✳ ❚❤❡ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ ❛ r❛♥❞♦♠ ✇❛❧❦ t♦ ❜❡ ❛ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ❛r❡ ❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✹✳ ❚❤❡ ❝♦♥❝❧✉s✐♦♥ ✐s ❞r❛✇♥ ✐♥ ❙❡❝t✐♦♥ ✺✳

✷ ▼♦❞❡❧

❈♦♥s✐❞❡r ❛ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ r❛♥❞♦♠ ✇❛❧❦ P ♦♥ t❤❡ ♣❛✐rs S = {(i, j), i, j ∈

N_{0} ♦❢ ♥♦♥✲♥❡❣❛t✐✈❡ ✐♥t❡❣❡rs✳ ❲❡ r❡❢❡r t♦ {(i, j)|i > 0, j > 0}✱ {(i, j)|i >}

0, j = 0}✱ {(i, j)|i = 0, j > 0} ❛♥❞ (0, 0) ❛s t❤❡ ✐♥t❡r✐♦r✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s✱ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s ❛♥❞ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ st❛t❡ s♣❛❝❡✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② ❢r♦♠ st❛t❡ (i, j) t♦ st❛t❡ (i + s, j + t) ✐s ❞❡♥♦t❡❞ ❜② ps,t(i, j)✳ ❚r❛♥s✐t✐♦♥s ❛r❡ r❡str✐❝t❡❞ t♦ t❤❡ ❛❞❥♦✐♥✐♥❣ ♣♦✐♥ts ✭❤♦r✐③♦♥t❛❧❧②✱ ✈❡rt✐❝❛❧❧② ❛♥❞ ❞✐❛❣♦♥❛❧❧②✮✱ ✐✳❡✳✱ ps,t(k, l) = 0✐❢ |s| > 1 ♦r |t| > 1✳ ❚❤❡ ♣r♦❝❡ss ✐s ❤♦♠♦❣❡♥❡♦✉s ✐♥ t❤❡ s❡♥s❡ t❤❛t ❢♦r ❡❛❝❤ ♣❛✐r (i, j)✱ (k, l) ✐♥ t❤❡ ✐♥t❡r✐♦r ✭r❡s♣❡❝t✐✈❡❧② ♦♥ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❞ ♦♥ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s✮ ♦❢ t❤❡ st❛t❡ s♣❛❝❡ ps,t(i, j) = ps,t(k, l) ❛♥❞ ps,t(i − s, j − t) = ps,t(k − s, l − t), ✭✷✮ ❢♦r ❛❧❧ −1 ≤ s ≤ 1 ❛♥❞ −1 ≤ t ≤ 1✳ ❲❡ ✐♥tr♦❞✉❝❡✱ ❢♦r i > 0✱ j > 0✱ t❤❡ ♥♦t❛t✐♦♥ ps,t(i, j) = ps,t✱ ps,0(i, 0) = hs ❛♥❞ p0,t(0, j) = vt✳ ◆♦t❡ t❤❛t t❤❡ ✜rst ❡q✉❛❧✐t② ♦❢ ✭✷✮ ✐♠♣❧✐❡s t❤❛t t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❢♦r ❡❛❝❤ ♣❛rt ♦❢ ✸

→i ↑j h1 p1,1 v1 h−1 h1 p1,1 p0,1 p−1,1 v−1 p1,0 v1 p1,−1 p1,1 p1,0 p1,1 p0,1 p−1,1 p−1,0 p−1,−1 p0,−1 p1,−1 1−h1−v1−p1,1 h0 p0,0 v0 ❋✐❣✉r❡ ✶✿ ❘❛♥❞♦♠ ✇❛❧❦ ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡✳ t❤❡ st❛t❡ s♣❛❝❡ ❛r❡ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✳ ❚❤❡ s❡❝♦♥❞ ❡q✉❛❧✐t② ❡♥s✉r❡s t❤❛t ❛❧s♦ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ❡♥t❡r✐♥❣ t❤❡ s❛♠❡ ♣❛rt ♦❢ t❤❡ st❛t❡ s♣❛❝❡ ❛r❡ tr❛♥s❧❛t✐♦♥ ✐♥✈❛r✐❛♥t✳ ❚❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥s ✐♠♣❧② t❤❛t p1,0(0, 0) = h1 ❛♥❞ p0,1(0, 0) = v1✳ ❚❤❡ ♠♦❞❡❧ ❛♥❞ ♥♦t❛t✐♦♥s ❛r❡ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✶✳ ❲❡ ❛ss✉♠❡ t❤❛t t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐s ❡r❣♦❞✐❝✱✐✳❡✳✱ ✐rr❡❞✉❝✐❜❧❡✱ ❛♣❡r✐♦❞✐❝ ❛♥❞ ♣♦s✐t✐✈❡ r❡❝✉rr❡♥t✱ ❛♥❞ ❤❛s ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ m✱ ✐✳❡✳✱ ❢♦r i, j > 0✱ m(i, j) = 1 X s=−1 1 X t=−1 m(i − s, j − t)ps,t, ✭✸✮ m(i, 0) = 1 X s=−1 m(i − s, 1)ps,−1+ 1 X s=−1 m(i − s, 0)ps,0, m(0, j) = 1 X t=−1 m(1, j − t)p−1,t+ 1 X t=−1 m(0, j − t)p0,t. ❚❤❡ ❜❛❧❛♥❝❡ ❛t t❤❡ ♦r✐❣✐♥ ✐s ✐♠♣❧✐❡❞ ❜② t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❢♦r ❛❧❧ ♦t❤❡r st❛t❡s✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥s ❛s t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r✱ t❤❡ ❤♦r✐③♦♥t❛❧ ❛①✐s ❛♥❞ t❤❡ ✈❡rt✐❝❛❧ ❛①✐s ♦❢ t❤❡ st❛t❡ s♣❛❝❡ r❡s♣❡❝✲ t✐✈❡❧②✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♠❡❛s✉r❡s t❤❛t ❛r❡ t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ♣♦s✐t✐✈❡ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ❉❡✜♥✐t✐♦♥ ✶ ✭■♥❞✉❝❡❞ ♠❡❛s✉r❡✮✳ ❚❤❡ ♠❡❛s✉r❡ m ✐s ❝❛❧❧❡❞ ✐♥❞✉❝❡❞ ❜② Γ ⊂ ✹

0 0.5 1 1.5 0.5 1 1.5 ✭❛✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❜✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❝✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❞✮ ❋✐❣✉r❡ ✷✿ ❊①❛♠♣❧❡s ♦❢ Q(x, y) = 0✳ ✭❛✮ p1,0 = p0,1 = 1₅✱ p−1,−1 = 3₅✳ ✭❜✮ p1,0 = 1₅✱ p0,−1 = p−1,1 = 2₅✳ ✭❝✮ p1,1 = ₆₂1, p−1,1 = p1,−1 = 10₃₁, p−1,−1 = 21₆₂✳ ✭❞✮ p−1,1= p1,−1= 1₄, p−1,−1= 1₂✳ R2₊ ✐❢ m(i, j) = X (ρ,σ)∈Γ α(ρ, σ)ρiσj, ✇✐t❤ α(ρ, σ) ∈ R\{0} ❢♦r ❛❧❧ (ρ, σ) ∈ Γ✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ✜♥✐t❡ ♠❡❛s✉r❡s✱ ✐♥ ✇❤✐❝❤ t❤❡ s✉♠ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♦r❞❡r✐♥❣ ♦❢ t❤❡ t❡r♠s✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❛ss✉♠❡ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡✱ X (ρ,σ)∈Γ |α(ρ, σ)| 1 1 − ρ 1 1 − σ < ∞. ✭✹✮ ❚♦ ✐❞❡♥t✐❢② t❤❡ ❣❡♦♠❡tr✐❝ ♠❡❛s✉r❡s t❤❛t ✐♥❞✐✈✐❞✉❛❧❧② s❛t✐s❢② t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✱ ✭✸✮✱ ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♣♦❧②♥♦♠✐❛❧ Q(x, y) = xy 1 X s=−1 1 X t=−1 x−sy−tps,t− 1 ! , t♦ ❝❛♣t✉r❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❜❛❧❛♥❝❡✱ ✐✳❡✳✱ Q(ρ, σ) = 0 ✐♠♣❧✐❡s t❤❛t m(i, j) = ρi_σj_{, i, j ∈ S s❛t✐s✜❡s ✭✸✮✳ ❙❡✈❡r❛❧ ❡①❛♠♣❧❡s ♦❢ t❤❡ ❧❡✈❡❧ s❡t Q(x, y) = 0} ❛r❡ ❞✐s♣❧❛②❡❞ ✐♥ ❋✐❣✉r❡ ✷✳ ▲❡t Q ❞❡♥♦t❡ t❤❡ s❡t ♦❢ r❡❛❧ (x, y) s❛t✐s❢②✐♥❣ Q(x, y) = 0✱ ✐✳❡✳✱ Q =n_{(x, y) ∈ R}2 _{| Q(x, y) = 0}o. ✭✺✮ ❲❡ ❛r❡ ♦♥❧② ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q(x, y) = 0 ✐♥ R2 +✱ ✇❤❡r❡ R2+= {(x, y)|x ≥ 0, y ≥ 0}✳ ❲❡ ❞❡♥♦t❡ t❤❡ s✉❜s❡t ♦❢ Q(x, y) = 0 ✐♥ R2₊ ❜② Q+✳ ▼♦r❡♦✈❡r✱ ✇❡ ❞❡♥♦t❡ t❤❡ ❜♦✉♥❞❛r② ♦❢ R2₊ ❜② RB ✇❤❡r❡ RB = {(x, y) ∈ R2+|xy = 0}✳ ❉✉❡ t♦ t❤❡ r❡q✉✐r❡♠❡♥t ♦❢ ❛ ✜♥✐t❡ ♠❡❛s✉r❡✱ ✇❡ ✇✐❧❧ ❜❡ ✐♥t❡r❡st❡❞ ✐♥ U = {(x, y)|(x, y) ∈ (0, 1)2_{}✳ ■♥ ❛❞❞✐t✐♦♥ t♦ U✱ ✇❡} ✐♥tr♦❞✉❝❡ ¯U = [0, 1)2✳ ❚❤❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q(x, y) = 0 ✇✐❧❧ ❜❡ st✉❞✐❡❞ ✜rst ✐♥ ❙❡❝t✐♦♥ ✸✳ ✺

✭❛✮ ✭❜✮ ✭❝✮ ✭❞✮ ✭❡✮ ❋✐❣✉r❡ ✸✿ ❙✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s✿ ♥♦♥✲③❡r♦ tr❛♥s✐t✐♦♥s ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✳

✸ ❆❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ✐♥ R

2 ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ✐♥ R2_{✳ ❋❛②♦❧❧❡ ❡t} ❛❧✳ ❬✾❪ ❤❛✈❡ ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ t❤❛t ✐s ❞❡✜♥❡❞ t❤r♦✉❣❤ xy(P1 s=−1 P1 t=−1xsytps,t− 1) = 0 ❛♥❞ ❛r✐s❡s ❢r♦♠ st✉❞②✐♥❣ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✳ ❲❡ st✉❞② t❤❡ ♣♦❧②♥♦♠✐❛❧ Q(x, y) t❤❛t ❛r✐s❡s ♥❛t✉r❛❧❧② ❢r♦♠ t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥s✳ ■t ✐s ❝❧❡❛r t❤❛t t❤❡ ♣r♦♣❡rt✐❡s t❤❛t ❛r❡ ❞❡r✐✈❡❞ ✐♥ ❬✾❪ ❝❛♥ ❜❡ r❡❧❛t❡❞ t♦ t❤❡ ♣r♦♣❡rt✐❡s ♦❢ Q(x, y) ❜② ❝♦♥✲ s✐❞❡r✐♥❣ ˜pi,j = p−i,−j✳ ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✇✐❧❧ ♣r❡s❡♥t s♦♠❡ ♦❢ t❤❡ r❡s✉❧ts ❢r♦♠ ❬✾❪ t❤❛t ✇✐❧❧ ❜❡ ✉s❡❢✉❧ ✐♥ t❤❡ s❡q✉❡❧ ❛s ✇❡❧❧ ❛s ❛ ♥✉♠❜❡r ♦❢ ♥❡✇ r❡s✉❧ts✳ ❚❤❡ r❡s✉❧ts ❢r♦♠ ❬✾❪ ❛r❡ ♠♦st❧② ❛❧❣❡❜r❛✐❝ ♦❢ ♥❛t✉r❡✳ ❚❤❡ ♥❡✇ r❡s✉❧ts t❤❛t ✇❡ ♣r❡s❡♥t ❞❡❛❧ ✇✐t❤ t❤❡ ❣❡♦♠❡tr② ♦❢ Q✳ ❋✐rst✱ ❢♦❧❧♦✇✐♥❣ ❬✾❪ ✇❡ ❞❡✜♥❡ t❤❡ ♥♦t✐♦♥ ♦❢ ❛ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦✳ ❉❡✜♥✐t✐♦♥ ✷ ✭❙✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦✮✳ ❘❛♥❞♦♠ ✇❛❧❦ P ✐s ❝❛❧❧❡❞ s✐♥❣✉❧❛r ✐❢ t❤❡ ❛ss♦❝✐❛t❡❞ ♣♦❧②♥♦♠✐❛❧ Q(x, y) ✐s ❡✐t❤❡r r❡❞✉❝✐❜❧❡ ♦r ♦❢ ❞❡❣r❡❡ 1 ✐♥ ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✳ ■♥ t❤❡ r❡♠❛✐♥❞❡r ✇❡ ✇✐❧❧ ♦❢t❡♥ ❞✐st✐♥❣✉✐s❤ ❜❡t✇❡❡♥ s✐♥❣✉❧❛r ❛♥❞ ♥♦♥✲ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ t❤❡ s✐♥❣✉❧❛r✐t✐❡s ♦❢ ❝✉r✈❡ Q✱ ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ✐♥ t❤❡ ✉s✉❛❧ ✇❛② ❜❡❧♦✇✳ ◆♦t❡ t❤❛t t❤❡ t✇♦ ♥♦t✐♦♥s ❛r❡ ♥♦t r❡❧❛t❡❞❀ ✐t ✐s✱ ❢♦r ✐♥st❛♥❝❡✱ ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ ❢♦r ✇❤✐❝❤ Q ❤❛s ❛ s✐♥❣✉❧❛r✐t②✳ ❉❡✜♥✐t✐♦♥ ✸ ✭❙✐♥❣✉❧❛r✐t② ♦❢ Q✮✳ P♦✐♥t (x, y) ∈ Q ✐s ❛ s✐♥❣✉❧❛r✐t② ♦❢ ♠✉❧t✐✲ ♣❧✐❝✐t② m✱ m > 1✱ ✐✛ ❛t (x, y) ❛❧❧ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡s ♦❢ Q(x, y) ♦❢ ♦r❞❡r ❧❡ss t❤❛♥ m ✈❛♥✐s❤ ❛♥❞ ❛t ❧❡❛st ♦♥❡ ♣❛rt✐❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ ♦r❞❡r m ✐s ♥♦♥✲③❡r♦✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❝❤❛r❛❝t❡r✐③❡s s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s ✐♥ t❡r♠s ♦❢ t❤❡✐r tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s✳ ▲❡♠♠❛ ✶ ✭▲❡♠♠❛ 2.3.2 ❬✾❪✮✳ ❘❛♥❞♦♠ ✇❛❧❦ P ✐s s✐♥❣✉❧❛r ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞s✿ ✶✳ ❚❤❡r❡ ❡①✐sts (i, j) ∈ Z2_{✱ |i| ≤ 1✱ |j| ≤ 1✱ s✉❝❤ t❤❛t ♦♥❧② p} i,j ❛♥❞ p−i,−j ❛r❡ ❞✐✛❡r❡♥t ❢r♦♠ 0 ✭s❡❡ ❋✐❣✉r❡ ✸✭❛✮ ❛♥❞ t❤❡ ♦t❤❡r t❤r❡❡ ❝❛s❡s ♦❜t❛✐♥❡❞ ❜② r♦t❛t✐♦♥✮❀ ✻

✷✳ ❚❤❡r❡ ❡①✐sts i✱ |i| = 1✱ s✉❝❤ t❤❛t ❢♦r ❛♥② j✱ |j| ≤ 1✱ pi,j = 0 ✭s❡❡

❋✐❣✉r❡ ✸✭❜✮ ❛♥❞ ❋✐❣✉r❡ ✸✭❝✮✮❀

✸✳ ❚❤❡r❡ ❡①✐sts j✱ |j| = 1✱ s✉❝❤ t❤❛t ❢♦r ❛♥② i✱ |i| ≤ 1✱ pi,j = 0 ✭s❡❡

❋✐❣✉r❡ ✸✭❞✮ ❛♥❞ ❋✐❣✉r❡ ✸✭❡✮✮✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ r❡s✉❧ts t❤❛t ✇❡ ✉s❡ ❢r♦♠ ❬✾❪ ❛r❡ ❞❡✜♥❡❞ ✐♥ t❡r♠s ♦❢ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ t❤❡ ♠✉❧t✐✲✈❛❧✉❡❞ ❛❧❣❡❜r❛✐❝ ❢✉♥❝t✐♦♥s X(y) ❛♥❞ Y (x) ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ t❤r♦✉❣❤ Q(X(y), y) = Q(x, Y (x)) = 0, ❢♦r x, y ∈ C✳ ❋✐rst✱ ♦❜s❡r✈❡ t❤❛t ❜② r❡♦r❞❡r✐♥❣ t❤❡ t❡r♠s ✐♥ Q(x, y) = 0 ✇❡ ❣❡t ( 1 X s=−1 y−s+1p−1,s)x2+ ( 1 X s=−1 y−s+1p0,s− y)x + ( 1 X s=−1 y−s+1p1,s) = 0. ✭✻✮

❚❤❡r❡❢♦r❡✱ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ X(y) ❛r❡ t❤❡ r♦♦ts ♦❢ ∆x(y) = 0✇❤❡r❡ ∆x(y)

✐s ❞❡✜♥❡❞ ❛s ∆x(y) = ( 1 X s=−1 y−s+1p0,s− y)2− 4( 1 X s=−1 y−s+1p−1,s)( 1 X s=−1 y−s+1p1,s). ✭✼✮ ■♥ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ❜② r❡✇r✐t✐♥❣ Q(x, y) = 0 ✐♥t♦ ( 1 X t=−1 x−t+1pt,−1)y2+ ( 1 X t=−1 x−t+1pt,0− x)y + ( 1 X t=−1 x−t+1pt,1) = 0, ✭✽✮ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛r❡ t❤❡ r♦♦ts ♦❢ ∆y(x) = 0 ✇❤❡r❡ ∆y(x)✐s ❞❡✜♥❡❞ ❛s ∆y(x) = ( 1 X t=−1 x−t+1pt,0− x)2− 4( 1 X t=−1 x−t+1pt,−1)( 1 X t=−1 x−t+1pt,1). ✭✾✮ ◆❡①t✱ ✇❡ ♣r❡s❡♥t t✇♦ ❧❡♠♠❛s t❤❛t ❢✉❧❧② ❝❤❛r❛❝t❡r✐③❡s t❤❡ ❧♦❝❛t✐♦♥ ♦❢ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛♥❞ X(y) ✐♥ t❡r♠s ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦✳ ❚❤❡s❡ r❡s✉❧ts ♣r♦✈✐❞❡ ✉s ✇✐t❤ t❤❡ ♦♣♣♦rt✉♥✐t② t♦ ❝♦♥♥❡❝t t❤❡ ❣❡♦♠❡tr② ♦❢ Q ✇✐t❤ t❤❡ ✐♥t❡r✐♦r tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s✳ ❚❤❡ ✜rst ❧❡♠♠❛ ♣r❡s❡♥t❡❞ ❜❡❧♦✇ ❢♦❧❧♦✇s ❢r♦♠ ▲❡♠♠❛s 2.3.8✕2.3.10 ♦❢ ❬✾❪✳ ❚❤❡ r❡s✉❧t r❡❛❞✐❧② ❢♦❧❧♦✇s ✐❢ ♦♥❡ t❛❦❡s ✐♥t♦ ❛❝❝♦✉♥t t❤❛t ✐♥ t❤❡ ❝✉rr❡♥t ♣❛♣❡r ✇❡ ❝♦♥s✐❞❡r ♦♥❧② ❡r❣♦❞✐❝ r❛♥❞♦♠ ✇❛❧❦s✱ ✇❤❡r❡❛s ❬✾❪ ❛❧s♦ ❛❧❧♦✇s ❢♦r ♥♦♥✲❡r❣♦❞✐❝ r❛♥❞♦♠ ✇❛❧❦s✳ ▲❡♠♠❛ ✷ ✭▲❡♠♠❛ 2.3.8✕2.3.10 ❬✾❪✮✳ ❋♦r ❛❧❧ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s s✉❝❤ t❤❛t My 6= 0✱ Y (x) ❤❛s ❢♦✉r r❡❛❧ ❜r❛♥❝❤ ♣♦✐♥ts✳ ▼♦r❡♦✈❡r✱ Y (x) ❤❛s t✇♦ ❜r❛♥❝❤ ♣♦✐♥ts x1 ❛♥❞ x2 ✭r❡s♣✳ x3 ❛♥❞ x4✮ ✐♥s✐❞❡ ✭r❡s♣✳ ♦✉ts✐❞❡✮ t❤❡ ✉♥✐t ❝✐r❝❧❡✳ ✼

❋♦r t❤❡ ♣❛✐r (x3, x4)✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ ❤♦❧❞s✿ ✶✳ ✐❢ p−1,0> 2√p−1,−1p−1,1✱ t❤❡♥ x3 ❛♥❞ x4 ❛r❡ ♣♦s✐t✐✈❡❀ ✷✳ ✐❢ p−1,0 = 2√p−1,−1p−1,1✱ t❤❡♥ ♦♥❡ ♣♦✐♥t ✐s ✐♥✜♥✐t❡ ❛♥❞ t❤❡ ♦t❤❡r ✐s ♣♦s✐t✐✈❡✱ ♣♦ss✐❜❧② ✐♥✜♥✐t❡❀ ✸✳ ✐❢ p−1,0 < 2√p−1,−1p−1,1✱ t❤❡♥ ♦♥❡ ♣♦✐♥t ✐s ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ♦t❤❡r ✐s ♥❡❣❛t✐✈❡✳ ❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ ♣❛✐r (x1, x2)✱ ✶✳ ✐❢ p1,0> 2√p1,−1p1,1✱ t❤❡♥ x1 ❛♥❞ x2 ❛r❡ ♣♦s✐t✐✈❡❀ ✷✳ ✐❢ p1,0 = 2√p1,−1p1,1✱ t❤❡♥ ♦♥❡ ♣♦✐♥t ✐s 0 ❛♥❞ t❤❡ s❡❝♦♥❞ ✐s ♥♦♥✲ ♥❡❣❛t✐✈❡❀ ✸✳ ✐❢ p1,0 < 2√p1,−1p1,1✱ t❤❡♥ ♦♥❡ ♣♦✐♥t ✐s ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ♦t❤❡r ✐s ♥❡❣❛✲ t✐✈❡✳ ❋♦r ❛❧❧ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s ❢♦r ✇❤✐❝❤ My = 0✱ ♦♥❡ ♦❢ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ✐s ❡q✉❛❧ t♦ 1✳ ■♥ ❛❞❞✐t✐♦♥✱ ✶✳ ✐❢ Mx < 0✱ t❤❡♥ t✇♦ ♦t❤❡r ❜r❛♥❝❤ ♣♦✐♥ts ❤❛✈❡ ❛ ♠♦❞✉❧✉s ❜✐❣❣❡r t❤❛♥ 1 ❛♥❞ t❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡ ❤❛s ❛ ♠♦❞✉❧✉s ❧❡ss t❤❛♥ 1❀ ✷✳ ✐❢ Mx > 0✱ t❤❡♥ t✇♦ ❜r❛♥❝❤ ♣♦✐♥ts ❛r❡ ❧❡ss t❤❛♥ 1 ❛♥❞ t❤❡ ♠♦❞✉❧✉s ♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡ ✐s ❜✐❣❣❡r t❤❛♥ 1✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ♣♦s✐t✐✈✐t② ❝♦♥❞✐t✐♦♥s ❛r❡ t❤❡ s❛♠❡ ❛s t❤❡ ❝❛s❡ ✇❤❡♥ My 6= 0✳ ❚❤✐s ❧❡♠♠❛ ✐s tr✉❡ ❛❧s♦ ❢♦r X(y)✱ ✉♣ t♦ ❛ ♣r♦♣❡r s②♠♠❡tr✐❝ ❝❤❛♥❣❡ ♦❢ t❤❡ ♣❛r❛♠❡t❡rs✳ ❚❤❡ ♥❡①t ❧❡♠♠❛ ❞❡❛❧s ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts✳ ▲❡♠♠❛ ✸ ✭▲❡♠♠❛ 2.3.10 ❬✾❪✮✳ ❚❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ X(y) ❛♥❞ Y (x) ✇✐t❤ ♠✉❧t✐♣❧✐❝✐t② 2 ♦❝❝✉r ♦♥❧② ❛t 0✱ 1 ❛♥❞ ∞✳ ❚❤❡ ✜♥❛❧ r❡s✉❧t ♦❢ ❬✾❪ t❤❛t ✇❡ ♥❡❡❞ ❝❤❛r❛❝t❡r✐③❡s ❡r❣♦❞✐❝✐t② ♦❢ ❛ r❛♥❞♦♠ ✇❛❧❦ ✐♥ t❡r♠s ♦❢ t❤❡ ❞r✐❢t ✐♥ t❤❡ ✐♥t❡r✐♦r ❛♥❞ ❛❧♦♥❣ t❤❡ ❜♦✉♥❞❛r✐❡s ♦❢ t❤❡ st❛t❡ s♣❛❝❡✳ ▲❡t ✉s ✜rst ❞❡✜♥❡ t❤❡ ❞r✐❢t ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦ P ❛s M, M′ ❛♥❞ M′′✳ M = (Mx, My) = ( 1 X t=−1 p1,t− 1 X t=−1 p−1,t, 1 X s=−1 ps,1− 1 X s=−1 ps,−1), M′ = (M_x′, M_y′) = (h1+ p1,1− h−1− p−1,1, 1 X s=−1 ps,1), M′′= (M_x′′, M_y′′) = ( 1 X t=−1 p1,t, v1+ p1,1− v−1− p1,−1). ✽

❚❤❡♦r❡♠ ✶ ✭❚❤❡♦r❡♠ 1.2.1 ❬✾❪✮✳ ❈♦♥s✐❞❡r ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❛♣❡r✐♦❞✐❝ r❛♥❞♦♠ ✇❛❧❦ P ✱ ✇❤❡♥ M 6= 0✱ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐s ❡r❣♦❞✐❝ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❝♦♥❞✐t✐♦♥s ❤♦❧❞s✿ ✶✳ _     Mx< 0, My < 0, MxMy′ − MyMx′ < 0, MyMx′′− MxMy′′< 0; ✷✳ Mx< 0, My ≥ 0, MyMx′′− MxMy′′ < 0❀ ✸✳ Mx≥ 0, My < 0, MxMy′ − MyMx′ < 0✳ ■t ✇✐❧❧ ❜❡ s❤♦✇♥ t❤❛t ✇❤❡♥ M = 0✱ ✇❤✐❝❤ ♠❡❛♥s t❤❡ ❞r✐❢t ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐s ③❡r♦✱ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q+ _{r❡❞✉❝❡s t♦ ❛ s✐♥❣❧❡ ♣♦✐♥t (1, 1)✳ ❚❤❡r❡✲} ❢♦r❡✱ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❤❛✈❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦s ✇✐t❤ ③❡r♦ ❞r✐❢t✳ ❍❡♥❝❡✱ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ③❡r♦ ❞r✐❢t ✐s ♦❢ ♠✐♥♦r ✐♠♣♦rt❛♥❝❡ ✐♥ ♦✉r ♣❛♣❡r✳ ❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s s❡❝t✐♦♥ ✐s r❡✲ ❧❛t❡❞ t♦ ♥❡✇ r❡s✉❧ts ♦♥ t❤❡ ❣❡♦♠❡tr② ♦❢ Q✳ ❋✐rst✱ ✇❡ ✐♥✈❡st✐❣❛t❡ t❤❡ ♣♦ss✐❜❧❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ Q ❛♥❞ RB ✇❤❡r❡ RB ✐s t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✜rst q✉❛❞r❛♥t✳ ▲❡♠♠❛ ✹✳ ❈♦♥s✐❞❡r ❛ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ P ✳ ■❢ (x, y) ∈ Q ∩ R2 + t❤❡♥ ❡✐t❤❡r x > 0 ❛♥❞ y > 0 ♦r x = y = 0✱ ✐✳❡✳✱ Q ❝❛♥♥♦t ❝r♦ss RB ❡①❝❡♣t ✐♥ t❤❡ ♦r✐❣✐♥✳ Pr♦♦❢✳ ■❢ (x, y) ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ Q ❛♥❞ x = 0✱ t❤❡♥ y ♠✉st ❜❡ t❤❡ r♦♦t ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥✱ p1,−1y2+ p1,0y + p1,1 = 0. ✭✶✵✮ ❲❡ ♥♦✇ s❤♦✇ t❤❛t t❤❡ r♦♦ts ♦❢ ✭✶✵✮ ❛r❡ ♥♦♥✲♣♦s✐t✐✈❡ ❜② ❝♦♥s✐❞❡r✐♥❣ ❛❧❧ ♣♦s✲ s✐❜❧❡ ❝❤♦✐❝❡s ♦❢ p1,−1, p1,0 ❛♥❞ p1,1✳ ■❢ p1,−1 6= 0✱ t❤❡♥ ✭✶✵✮ ❤❛s ❡✐t❤❡r ♥♦ r♦♦t ♦r t✇♦ ♥♦♥✲♣♦s✐t✐✈❡ r♦♦ts ❜② ✐♥✈❡st✐❣❛t✐♥❣ t❤❡ r❡❧❛t✐♦♥s ♦❢ t❤❡ r♦♦ts ✉s✐♥❣ ❱✐❡t❛✬s ❢♦r♠✉❧❛s✳ ■❢ p1,−1 = 0 ❛♥❞ p1,0 6= 0✱ t❤❡♥ ✭✶✵✮ ❤❛s ♦♥❡ ♥♦♥✲♣♦s✐t✐✈❡ r♦♦t✳ ■❢ p1,−1 = p1,0 = 0 ❛♥❞ p1,1 6= 0✱ t❤❡♥ ✭✶✵✮ ❤❛s ♥♦ r♦♦t✳ ❚❤❡ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ p1,−1= p1,0= p1,1= 0 ✐s ❡①❝❧✉❞❡❞ ❜❡❝❛✉s❡ ✐t ✐s s✐♥❣✉❧❛r✳ ■♥ s✐♠✐✲ ❧❛r ❢❛s❤✐♦♥ ✐t ❢♦❧❧♦✇s t❤❛t Q(x, y) = 0 ❝❛♥ ♦♥❧② ✐♥t❡rs❡❝t y = 0 ✇❤❡♥ x ≤ 0✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ Q ❛♥❞ RB ✐s t❤❡ ♦r✐❣✐♥✳ ◆♦✇ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t✳ ▲❡♠♠❛ ✺✳ ❈♦♥s✐❞❡r ❛ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ P ✇✐t❤ ♥♦♥✲③❡r♦ ❞r✐❢t✳ ❚❤❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ❛r❡ ❝❧♦s❡❞✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ❤❛s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐♥ R2 +✳ ▼♦r❡♦✈❡r✱ t❤✐s ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ❤❛s ♥♦♥✲❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ t❤❡ ✉♥✐t sq✉❛r❡ U✳ ✾

Pr♦♦❢ ♦❢ ▲❡♠♠❛ ✺✳ ❋✐rst ♦❜s❡r✈❡ t❤❛t ❜❡❝❛✉s❡ ♦❢ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ❢✉♥❝t✐♦♥s ∆x(y) ❛♥❞ ∆y(x) ❞❡✜♥❡❞ ✐♥ ✭✼✮ ❛♥❞ ✭✾✮✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❝♦♥♥❡❝t❡❞ ❝♦♠✲ ♣♦♥❡♥ts ❛r❡ ❝❧♦s❡❞✳ ❚❤✐s ♠❡❛♥s✱ ❢♦r ❡❛❝❤ ❧✐♥❡ ✐♥ R2 _{t❤❛t t❤❡ ♠✉❧t✐♣❧✐❝✐t②} ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤✐s ❧✐♥❡ ❛♥❞ ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐s ❡✐t❤❡r t✇♦ ♦r ③❡r♦✳ ■t ❝❛♥ r❡❛❞✐❧② ❜❡ ✈❡r✐✜❡❞ t❤❛t (1, 1)✱ (1, P1s=−1ps,1 P1 s=−1ps,−1)❛♥❞ ( P1 t=−1p1,t P1 t=−1p−1,t , 1) ❛r❡ ♦♥ Q✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ❛t ❧❡❛st ♦♥❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t✳ ❚❤❡ ❡r❣♦❞✐❝ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ♥♦♥✲③❡r♦ ❞r✐❢t ❢r♦♠ ❚❤❡♦r❡♠ ✶ ✐♠♣❧② t❤❛t ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡q✉✐r❡♠❡♥ts ♠✉st ❜❡ s❛t✐s✜❡❞✱ 0 < P1 s=−1ps,1 P1 s=−1ps,−1 < 1, 0 < P1 t=−1p1,t P1 t=−1p−1,t < 1. ✭✶✶✮ ❲✐t❤♦✉t ❧♦ss ♦❢ ❣❡♥❡r❛❧✐t②✱ ❧❡t ✉s ❛ss✉♠❡ 0 < P1s=−1ps,1 P1 s=−1ps,−1 < 1✳ ■❢ (1, 1) ❛♥❞ (1, P1s=−1ps,1 P1 s=−1ps,−1) ❛r❡ ♦♥ ❞✐✛❡r❡♥t ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts✱ t❤❡♥ t❤❡s❡ t✇♦ ❝♦♠♣♦♥❡♥ts ♠✉st ❜❡ ❝♦♠♣❧❡t❡❧② ❝♦♥t❛✐♥❡❞ ✐♥ x ≤ 1 ❛♥❞ x ≥ 1 r❡s♣❡❝t✐✈❡❧②✱ ❞✉❡ t♦ t❤❡ ❝❧♦s❡❞♥❡ss ♦❢ t❤❡ ❝♦♠♣♦♥❡♥t ❛♥❞ ❞❡❣r❡❡ ♦❢ Q✳ ❍♦✇❡✈❡r✱ ✐♥ t❤❛t ❝❛s❡✱ t✇♦ ♦❢ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛r❡ ❡q✉❛❧ t♦ 1 ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ▲❡♠♠❛ ✷✳ ❚❤❡r❡❢♦r❡✱ (1, 1) ❛♥❞ (1, P1s=−1ps,1 P1 s=−1ps,−1) ❛r❡ ♦♥ t❤❡ s❛♠❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✳ ❲❡ ❤❛✈❡ ∆y(1) 6= 0✱ ❞✉❡ t♦ t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ∆y(x)✱ t❤❡ ❝♦♥♥❡❝t❡❞ ❝♦♠✲ ♣♦♥❡♥t ❝♦♥t❛✐♥✐♥❣ (1, 1) ♠✉st ❡①t❡♥❞ t♦ ❜♦t❤ t❤❡ ✐♥s✐❞❡ ❛♥❞ ♦✉ts✐❞❡ ♦❢ t❤❡ ✉♥✐t sq✉❛r❡ U✳ ■❢ Q ❝♦♥t❛✐♥s ♦♥❧② ♦♥❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✱ t❤✐s ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♠✉st ❜❡ ✐♥ R2 + ❜❡❝❛✉s❡ Q ❝❛♥ ♦♥❧② ✐♥t❡rs❡❝t RB ❛t t❤❡ ♦r✐❣✐♥✳ ■t st✐❧❧ r❡♠❛✐♥s t♦ s❤♦✇ t❤❛t Q ❝♦♥t❛✐♥s ❛ s✐♥❣❧❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐♥ R2₊✳ ■❢ Q ✇♦✉❧❞ ❝♦♥t❛✐♥ ♠✉❧t✐♣❧❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts ✐♥ R2₊✱ ❜② ✉s✐♥❣ t❤❡ ♣r♦♣❡rt② t❤❛t t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥s ♦❢ ❛♥② ❤♦r✐③♦♥t❛❧ ♦r ✈❡rt✐❝❛❧ ❧✐♥❡ ❛♥❞ Q(x, y) = 0 ✐s ❛t ♠♦st t✇♦✱ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❡r❡ ❛r❡ t❤r❡❡ ❜r❛♥❝❤ ♣♦✐♥ts ❡✐t❤❡r ✐♥s✐❞❡ ♦r ♦✉ts✐❞❡ t❤❡ ✉♥✐t sq✉❛r❡✱ ✇❤✐❝❤ ❝♦♥tr❛❞✐❝ts ▲❡♠♠❛ ✷✳ ❚❤❡r❡❢♦r❡✱ t❤❡r❡ ✐s ♦♥❡ ❛♥❞ ♦♥❧② ♦♥❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ Q ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ❛♥❞ ✐t ❤❛s ♥♦♥✲❡♠♣t② ✐♥t❡rs❡❝t✐♦♥ ✇✐t❤ U✳ ❘❡♠❛r❦✿ ❲❤❡♥ t❤❡ ❞r✐❢t ♦❢ t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐s ③❡r♦✱ t❤❡ ✉♥✐q✉❡ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t r❡❞✉❝❡s t♦ ♣♦✐♥t (1, 1)✱ s❡❡ ❙❡❝t✐♦♥ 6.5 ❬✾❪✳ ◆♦✇ ✇❡ ❦♥♦✇ t❤❛t Q+_{✱ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ Q(x, y) = 0 ❛♥❞ R}2 +✱ ✐s ❛ ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✳ ❉❡♥♦t❡ t❤❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛♥❞ X(y) ♦♥ Q+ ❜② xl, xr ✇✐t❤ xl < xr ❛♥❞ yb, yt ✇✐t❤ yb < yt r❡s♣❡❝t✐✈❡❧②✳ ▲❡t yl, yr, xb, xt s❛t✐s❢②(xl, yl), (xr, yr), (xt, yt), (xb, yb) ∈ Q✱ s❡❡ ❋✐❣✉r❡ ✹✭❛✮✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ (xl, yl), (xr, yr), (xt, yt), (xb, yb) ❛s ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Q+✳ ❋r♦♠ ▲❡♠♠❛ ✷✱ ✇❡ ❦♥♦✇ t❤❛t 0 ≤ xl ≤ 1 ≤ xr✱ 0 ≤ yb ≤ 1 ≤ yt✳ ❙✐♥❝❡ ✇❡ ❛r❡ ♦♥❧② ✐♥t❡r❡st❡❞ ✐♥ ✜♥✐t❡ ♠❡❛s✉r❡s✱ ✇❡ ♦♥❧② ❝♦♥s✐❞❡r Q+ _{✐♥ ¯U✳ ❘❡❝❛❧❧ ❢r♦♠ ❙❡❝t✐♦♥ ✷ t❤❛t} ✶✵

0 0.5 1 1.4 0.5 1 1.4 (xl,yl) (xb,yb) (xt,yt) (xr,yr) ✭❛✮ 0 0.5 1 1.4 0.5 1 1.4 Q00 Q10 Q11 Q01 ✭❜✮ ❋✐❣✉r❡ ✹✿ Q+ _{❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ ❢r♦♠ ❋✐❣✉r❡ ✷✭❝✮✳ ✭❛✮ ❇r❛♥❝❤ ♣♦✐♥ts ♦❢} Q+✳ ✭❜✮ P❛rt✐t✐♦♥ ♦❢ Q+✳ ¯ U = [0, 1)2_{✳ ▲❡♠♠❛ ✺ st❛t❡s t❤❛t Q}+ U = Q+∩ U ✐s ❛ ♥♦♥✲❡♠♣t② s❡t ❢♦r ❛♥ ❡r❣♦❞✐❝ r❛♥❞♦♠ ✇❛❧❦ ✇✐t❤ ♥♦♥✲③❡r♦ ❞r✐❢t✳ ❲❡ ✜rst st❛rt t❤❡ ❛♥❛❧②s✐s ♦❢ Q+_✳ ❉❡✜♥✐t✐♦♥ ✹ ✭P❛rt✐t✐♦♥ ♦❢ Q+_{✮✳ ❚❤❡ ♣❛rt✐t✐♦♥ {Q} 00, Q01, Q10, Q11} ♦❢ Q+ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ Q00 ✐s t❤❡ ♣❛rt ♦❢ Q ❝♦♥♥❡❝t✐♥❣ (xl, yl) ❛♥❞ (xb, yb)❀ Q10 ✐s t❤❡ ♣❛rt ♦❢ Q ❝♦♥♥❡❝t✐♥❣ (xb, yb) ❛♥❞ (xr, yr)❀ Q01 ✐s t❤❡ ♣❛rt ♦❢ Q ❝♦♥♥❡❝t✐♥❣ (xl, yl) ❛♥❞ (xt, yt)❀ Q11 ✐s t❤❡ ♣❛rt ♦❢ Q ❝♦♥♥❡❝t✐♥❣ (xr, yr) ❛♥❞ (xt, yt)✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ Q+ _{✐s ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✹✭❜✮✳ ❚❤❡} ♠♦♥♦t♦♥✐❝✐t② ♦❢ X(y) ❛♥❞ Y (x) ♣❧❛②s ❛♥ ❝r✉❝✐❛❧ r♦❧❡ ✐♥ ❛♥❛❧②③✐♥❣ t❤❡ ♣❛✐r✇✐s❡✲ ❝♦✉♣❧❡ s❡t ✇✐t❤ ✐♥✜♥✐t❡ ❝❛r❞✐♥❛❧✐t②✱ t❤❡r❡❢♦r❡ ✇❡ ❛♥❛❧②③❡ ✐t ❤❡r❡✳

▲❡♠♠❛ ✻✳ ❈♦♥s✐❞❡r (x, y) ∈ Qi,1−i ❛♥❞ (˜x, ˜y) ∈ Qi,1−i ✇❤❡r❡ i = 0, 1✱ ✐❢

˜

x > x✱ t❤❡♥ ˜y > y✳ ❈♦♥s✐❞❡r (x, y) ∈ Qi,i ❛♥❞ (˜x, ˜y) ∈ Qi,i ✇❤❡r❡ i = 0, 1✱ ✐❢

˜

x > x✱ t❤❡♥ ˜y < y✳

Pr♦♦❢✳ ❲❡ ♦♥❧② ❝♦♥s✐❞❡r Q01✳ ❚❤❡ ♣r♦♦❢s ❢♦r t❤❡ ♦t❤❡r ❝❛s❡s ❢♦❧❧♦✇ ❛♥❛❧♦✲

❣♦✉s❧②✳ ❆ss✉♠❡ (x, y), (˜x, ˜y) ∈ Q01 ❛♥❞ ˜x > x✱ ✐❢ ˜y ≤ y✱ t❤❡♥ t❤❡r❡ ❡①✐sts

(ρ, σ), (˜ρ, ˜_{σ) ∈ Q}01 s✉❝❤ t❤❛t ˜σ = σ✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ❛❧s♦ ❡①✐sts ❛♥♦t❤❡r ❣❡♦♠❡tr✐❝ t❡r♠ (ˆρ, ˆσ) ∈ Q10∪ Q11 s✉❝❤ t❤❛t ˆσ = σ✳ ❚❤✐s ❝♦♥tr❛❞✐❝ts t❤❛t t❤❡ ♠✉❧t✐♣❧✐❝✐t② ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛♥② ❤♦r✐③♦♥t❛❧ ❧✐♥❡ ❛♥❞ Q ✐s ❛t ♠♦st t✇♦✱ ✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳ ◆❡①t✱ ✇❡ t✉r♥ ♦✉r ❛tt❡♥t✐♦♥ t♦ t❤❡ s✐♥❣✉❧❛r✐t② ♦❢ Q✱ ❛s ❞❡✜♥❡❞ ✐♥ ❉❡✜♥✐✲ t✐♦♥ ✸✳ ❲❡ ✇✐❧❧ s❡❡ ❜❡❧♦✇ t❤❛t ✐❢ ♦♥❡ s❡t ❢r♦♠ t❤❡ ♣❛rt✐t✐♦♥ ♦❢ Q+ _{✐s ❡♠♣t②✱} t❤❡♥ t❤❡ ❝✉r✈❡ Q+_{✇✐❧❧ ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t②✳ ❚❤❡ s✐♥❣✉❧❛r✐t② ♣❧❛②s ❛♥ ✐♠♣♦rt❛♥t} r♦❧❡ ✐♥ t❤❡ ❛♥❛❧②s✐s ❧❛t❡r✳ ▲❡♠♠❛ ✼✳ ❋♦r ❛❧❧ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s ✇✐t❤ ♥♦♥✲③❡r♦ ❞r✐❢t✱ (x, y) ✐s ❛ s✐♥❣✉❧❛r✐t② ♦❢ Q+ _{✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ❝r✉♥♦❞❡ ♦❢ ♦r❞❡r ✷ ❛♥❞ x ❛♥❞ y ❛r❡} ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ ♠✉❧t✐♣❧✐❝✐t② 2 ♦❢ Y (x) ❛♥❞ X(y) r❡s♣❡❝t✐✈❡❧②✳ ✶✶

−0.5 −0.25 0 0.25 0.5 −0.5 −0.25 0 0.25 0.5 ✭❛✮ −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 ✭❜✮ −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 ✭❝✮ ❋✐❣✉r❡ ✺✿ ❚②♣❡s ♦❢ r❡❛❧ ❞♦✉❜❧❡ ♣♦✐♥t✳ ✭❛✮ ❝r✉♥♦❞❡✳ ✭❜✮ ❛❝♥♦❞❡✳ ✭❝✮ ♦r❞✐♥❛r② ❝✉s♣✳ Pr♦♦❢✳ ❲❡ ♣r♦✈❡ ❜② ❝♦♥tr❛❞✐❝t✐♦♥ t❤❛t ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ s✐♥❣✉❧❛r✐t② ♦❢ ♦r❞❡r ❧❛r❣❡r t❤❛♥ 2✳ ❙✉♣♣♦s❡ t❤❛t (˜x, ˜y) ∈ Q+ _{✐s ❛ s✐♥❣✉❧❛r✐t② ♦❢ ♦r❞❡r} ❧❛r❣❡r t❤❛♥ 2✳ ❋r♦♠ ▲❡♠♠❛ ✹ ✐t ❢♦❧❧♦✇s t❤❛t ✇❡ ♥❡❡❞ t♦ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡s ✐✮ ˜x > 0 ❛♥❞ ˜y > 0✱ ❛♥❞ ✐✐✮ (˜x, ˜y) = (0, 0)✳ ■❢ ˜x > 0 ❛♥❞ ˜y > 0 ✐t ❢♦❧❧♦✇s ❢r♦♠

∂2_{Q(x, y)} ∂2_x = 1 X t=−1 p−1,ty−t+1 = 0, ❛♥❞ ∂2_{Q(x, y)} ∂2_y = 1 X s=−1 ps,−1x−s+1= 0, t❤❛t p−1,1 = p−1,0 = p−1,−1 = p0,−1 = p1,−1 = 0✱ ✇❤✐❝❤ ❧❡❛❞s t♦ ❛ ♥♦♥✲ ❡r❣♦❞✐❝ r❛♥❞♦♠ ✇❛❧❦✳ ❋♦r (˜x, ˜y) = (0, 0) ✐t ❢♦❧❧♦✇s ❢r♦♠ ∂2_{Q(x, y)}

∂x∂y = 4xyp−1,−1+ 2xp−1,0+ 2yp0,−1+ p0,0− 1 = 0,

t❤❛t p00 = 1✱ ✇❤✐❝❤ ❧❡❛❞s t♦ ❛ r❛♥❞♦♠ ✇❛❧❦ t❤❛t ✐s ♥♦t ✐rr❡❞✉❝✐❜❧❡✳ ❚❤✐s ❝♦♥❝❧✉❞❡s t❤❡ ♣r♦♦❢ t❤❛t ❛ s✐♥❣✉❧❛r✐t② ❤❛s ❛t ♠♦st ♦r❞❡r 2✳ ◆❡①t✱ ✇❡ ❞❡♠♦♥str❛t❡ t❤❛t ✐❢ (x, y) ✐s ❛ s✐♥❣✉❧❛r✐t②✱ t❤❡♥ x ❛♥❞ y ❛r❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛♥❞ X(y) r❡s♣❡❝t✐✈❡❧②✳ ❇② ❝♦♠❜✐♥✐♥❣ Q(x, y) = 0 ✇✐t❤ ∂Q(x, y) ∂x = 2x( 1 X t=−1 p−1,ty−t+1) + ( 1 X t=−1 p0,ty−t+1− y) = 0 ✇❡ ♦❜t❛✐♥ 1 X t=−1 p−1,ty−t+1x2= 1 X t=−1 p1,ty−t+1, ✇❤✐❝❤ ♠❡❛♥s x ✐s t❤❡ r♦♦t ♦❢ ∆y(x) = 0✱ ❞❡✜♥❡❞ ✐♥ ✭✾✮ ❛♥❞ t❤❡r❡❢♦r❡ ❛ ❜r❛♥❝❤ ♣♦✐♥t ♦❢ Y (x)✳ ❙✐♠✐❧❛r❧②✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ Q(x, y) = 0 ❛♥❞ ∂Q(x, y)/∂y = 0 t❤❛t y ✐s ❛ ❜r❛♥❝❤ ♣♦✐♥t ♦❢ X(y)✳ ◆♦✇✱ ✇❡ ❛r❡ r❡❛❞② t♦ ♣r♦✈❡ t❤❛t ❛ s✐♥❣✉❧❛r✐t② (x, y) ✐s ❛ ❝r✉♥♦❞❡✳ ❲❡ r❡❢❡r t❤❡ r❡❛❞❡r t♦ ❛♥② s♦✉r❝❡ ♦♥ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡s ❢♦r ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ♦♥ s✐♥❣✉❧❛r✐t✐❡s✱ ❢♦r ✐♥st❛♥❝❡ ❬✶✵❪✳ ❆♥ ✐❧❧✉str❛t✐♦♥ ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ s✐♥❣✉❧❛r✐t✐❡s ✶✷

♦❢ ♦r❞❡r 2 ✐s ❣✐✈❡♥ ✐♥ ❋✐❣✉r❡ ✺✳ ◆♦t❡✱ t❤❛t t❤❡ ✜❣✉r❡ ❞♦❡s ♥♦t ✐♥❝❧✉❞❡ ❛ r❛♠♣❤♦✐❞ ❝✉s♣✱ s✐♥❝❡ ✐t ❤❛s ♦r❞❡r ❧❛r❣❡r t❤❛♥ 2✳ ❆ s✐♥❣✉❧❛r✐t② ❝❛♥♥♦t ❜❡ ❛♥ ♦r❞✐♥❛r② ❝✉s♣✱ ❜❡❝❛✉s❡ x ❛♥❞ y ❛r❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛♥❞ X(y) r❡s♣❡❝t✐✈❡❧②✳ ▼♦r❡♦✈❡r✱ (x, y) ✐s ♥♦t ❛♥ ❛❝♥♦❞❡ ❜❡❝❛✉s❡ Q+ U ✐s ♥♦♥✲❡♠♣t② ❞✉❡ t♦ ▲❡♠♠❛ ✺✳ ❚❤❡r❡❢♦r❡✱ ❛ s✐♥❣✉❧❛r✐t② ✐s ❛ ❝r✉♥♦❞❡✳ ❚❤❡ ✜♥❛❧ r❡s✉❧t ✐♥ t❤✐s ❧❡♠♠❛ ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ♦❜s❡r✈❛t✐♦♥ t❤❛t ✐❢ x ❛♥❞ y ❛r❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ Y (x) ❛♥❞ X(y) r❡s♣❡❝t✐✈❡❧② ❛♥❞ (x, y) ✐s ❛ ❝r✉♥♦❞❡ t❤❡♥ x ❛♥❞ y ♠✉st ❤❛✈❡ ♠✉❧t✐♣❧✐❝✐t② t✇♦✳ ▲❡♠♠❛ ✽✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ❤❛s ❛ s✐♥❣✉❧❛r✐t② ✐♥ ¯U ✐❢ ❛♥❞ ♦♥❧② ✐❢ p0,1= p1,1 = p1,0 = 0✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ✐t ✐s ❧♦❝❛t❡❞ ✐♥ t❤❡ ♦r✐❣✐♥✳ Pr♦♦❢✳ ▲❡♠♠❛ ✼ st❛t❡s t❤❛t (x, y) ✐s ❛ s✐♥❣✉❧❛r✐t② ♦❢ Q+ _{✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛} ❝r✉♥♦❞❡ ♦❢ ♦r❞❡r ✷ ❛♥❞ x ❛♥❞ y ❛r❡ ❜r❛♥❝❤ ♣♦✐♥ts ♦❢ ♠✉❧t✐♣❧✐❝✐t② 2 ♦❢ Y (x) ❛♥❞ X(y) r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ♦♥❧② ♥❡❡❞ t♦ ❝♦♥s✐❞❡r (x, y) ✇❤❡r❡ x ❛♥❞ y ❛r❡ t❤❡ ♠✉❧t✐♣❧❡ r♦♦ts ♦❢ ∆y(x) = 0 ❛♥❞ ∆x(y) = 0 r❡s♣❡❝t✐✈❡❧②✳ ❆ ♠✉❧t✐♣❧❡ r♦♦t ♦❢ ∆y(x) = 0 ❛♥❞ ∆x(y) = 0 ❝❛♥ ♦♥❧② ♦❝❝✉r ❛t x = 0, 1 ♦r ∞ ❛♥❞ y = 0, 1 ♦r ∞✱ r❡s♣❡❝t✐✈❡❧②✱ ❞✉❡ t♦ ▲❡♠♠❛ ✸✳ ❚❤❡r❡❢♦r❡✱ x = 0 ❛♥❞ y = 0 ♠✉st ❜❡ ♠✉❧t✐♣❧❡ r♦♦ts ♦❢ ∆y(x) = 0 ❛♥❞ ∆x(y) = 0✱ r❡s♣❡❝t✐✈❡❧②✱ ✐❢ t❤❡r❡ ✐s ❛ s✐♥❣✉❧❛r✐t② ✐♥ ¯U✳ ❲❡ ❦♥♦✇ ❢r♦♠ ❬✾✱ ▲❡♠♠❛ 2.3.10❪ t❤❛t ∆y(x) = 0 ❤❛s t❤❡ ♠✉❧t✐♣❧❡ r♦♦t ❛t 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ p−1,0 = p−1,1= p0,1 = 0, ✭✶✷✮ p1,0 = p1,1 = p0,1 = 0, ✭✶✸✮ p−1,−1 = p0,−1 = p1,−1 = 0 ✭✶✹✮ ❛♥❞ ∆x(y) = 0❤❛s t❤❡ ♠✉❧t✐♣❧❡ r♦♦t ❛t 0 ✐❢ ❛♥❞ ♦♥❧② ✐❢ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞s✿ p0,−1 = p1,−1= p1,0 = 0, ✭✶✺✮ p0,1 = p1,1 = p1,0 = 0, ✭✶✻✮ p−1,−1 = p−1,0= p−1,1= 0. ✭✶✼✮ ❈♦♥❞✐t✐♦♥s ✭✶✹✮ ❛♥❞ ✭✶✼✮ ❛r❡ ❡①❝❧✉❞❡❞ ❜❡❝❛✉s❡ ❡✐t❤❡r ♦❢ t❤❡♠ ✇✐❧❧ ❧❡❛❞ t♦ ❛ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦✳ ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❝♦♥❞✐t✐♦♥s ✭✶✷✮ ❛♥❞ ✭✶✺✮✱ ✭✶✷✮ ❛♥❞ ✭✶✻✮✱ ✭✶✸✮ ❛♥❞ ✭✶✺✮ ✇✐❧❧ ❧❡❛❞ t♦ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s ❛s ✇❡❧❧✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ❤❛s ❛ s✐♥❣✉❧❛r✐t② ✐♥ ¯U ✐❢ ❛♥❞ ♦♥❧② ✐❢ p0,1 = p1,1 = p1,0 = 0✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ✐t ✐s ❧♦❝❛t❡❞ ✐♥ t❤❡ ♦r✐❣✐♥✳

✹ ❈♦♥str❛✐♥ts t♦ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡s ❛♥❞ r❛♥❞♦♠

✇❛❧❦s

■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ✜rst ❝❤❛r❛❝t❡r✐③❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❝❛♥❞✐❞❛t❡ s❡t ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ✇❤✐❝❤ ♠❛② ❧❡❛❞ t♦ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✳ ❚❤❡♥ ✇❡ ✇✐❧❧ ✶✸

0 0.5 1 1.4 0.5 1 1.4 ✭❛✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❜✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❝✮ 0 0.5 1 1.4 0.5 1 1.4 ✭❞✮ ❋✐❣✉r❡ ✻✿ P❛rt✐t✐♦♥s ♦❢ s❡t Γ✳ ✭❛✮ ❝✉r✈❡ Q+ _{♦❢ ❋✐❣✉r❡ ✷✭❞✮ ❛♥❞ Γ ⊂ Q}+ _❛s ❞♦ts✳ ✭❜✮ ❤♦r✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ✇✐t❤ 6 s❡ts✳ ✭❝✮ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉✲ ♣❧❡❞ ♣❛rt✐t✐♦♥ ✇✐t❤ 6 s❡ts✳ ✭❞✮ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ✇✐t❤ 4 s❡ts✳ ❉✐✛❡r❡♥t s❡ts ❛r❡ ♠❛r❦❡❞ ❜② ❞✐✛❡r❡♥t s②♠❜♦❧s✳ ♣r♦✈✐❞❡ ♥❡❝❡ss❛r② ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❛ r❛♥❞♦♠ ✇❛❧❦ t♦ ❛❧❧♦✇ ❢♦r ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s t♦ ❝♦♥st✐t✉t❡ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✳ ❋✐♥❛❧❧②✱ ✇❡ ❞❡♠♦♥str❛t❡ t❤❛t ✐t ✐s ♥❡❝❡ss❛r② t♦ ❤❛✈❡ ❛t ❧❡❛st ♦♥❡ ♥❡❣❛t✐✈❡ ❝♦❡✣❝✐❡♥t ✐♥ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ t❤❛t ✐s ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦✲ ♠❡tr✐❝ t❡r♠s✳ ✹✳✶ ❯♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥s ❚❤❡ ♣r♦♦❢s ✐♥ t❤✐s ❛♥❞ s✉❜s❡q✉❡♥t s❡❝t✐♦♥s ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♥♦t✐♦♥ ♦❢ ✉♥✲ ❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥s✱ ✇❤✐❝❤ ✐s ✐♥tr♦❞✉❝❡❞ ✜rst✳ ❉❡✜♥✐t✐♦♥ ✺ ✭❯♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥✮✳ ❆ ♣❛rt✐t✐♦♥ {Γ1, Γ2, · · · } ♦❢ Γ ✐s ❤♦r✲ ✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ✐❢ (ρ, σ) ∈ Γp ❛♥❞ (˜ρ, ˜σ) ∈ Γq ❢♦r p 6= q✱ ✐♠♣❧✐❡s t❤❛t ˜ ρ 6= ρ✱ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞ ✐❢ (ρ, σ) ∈ Γp ❛♥❞ (˜ρ, ˜σ) ∈ Γq ❢♦r p 6= q✱ ✐♠♣❧✐❡s ˜ σ 6= σ✱ ❛♥❞ ✉♥❝♦✉♣❧❡❞ ✐❢ ✐t ✐s ❜♦t❤ ❤♦r✐③♦♥t❛❧❧② ❛♥❞ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞✳ ❲❡ ❝❛❧❧ ❛ ♣❛rt✐t✐♦♥ ✇✐t❤ t❤❡ ❧❛r❣❡st ♥✉♠❜❡r ♦❢ s❡ts ❛ ♠❛①✐♠❛❧ ♣❛rt✐t✐♦♥✳ ▲❡♠♠❛ ✾ ✭▲❡♠♠❛ 1 ❬✻❪✮✳ ❚❤❡ ♠❛①✐♠❛❧ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥✱ t❤❡ ♠❛①✐♠❛❧ ❤♦r✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❛♥❞ t❤❡ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐✲ t✐♦♥ ❛r❡ ✉♥✐q✉❡✳ ❊①❛♠♣❧❡s ♦❢ ❛ ♠❛①✐♠❛❧ ❤♦r✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥✱ ♦❢ ❛ ♠❛①✐♠❛❧ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❛♥❞ ♦❢ ❛ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❋✐❣✉r❡ ✻✳ ▲❡t H ❞❡♥♦t❡ t❤❡ ♥✉♠❜❡r ♦❢ ❡❧❡♠❡♥ts ✐♥ t❤❡ ♠❛①✐♠❛❧ ❤♦r✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❛♥❞ Γh p✱ p = 1, . . . , H✱ t❤❡ s❡ts t❤❡♠s❡❧✈❡s✱ ✇❤❡r❡ ❡❧❡♠❡♥ts ♦❢ Γh p ❤❛✈❡ ❝♦♠♠♦♥ ❤♦r✐③♦♥t❛❧ ❝♦♦r❞✐♥❛t❡ ρ(Γhp)✳ ❚❤❡ ♠❛①✲ ✐♠❛❧ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❤❛s V s❡ts✱ Γv q✱ q = 1, · · · , V ✱ ✇❤❡r❡ ❡❧❡♠❡♥ts ♦❢ Γv q ❤❛✈❡ ❝♦♠♠♦♥ ✈❡rt✐❝❛❧ ❝♦♦r❞✐♥❛t❡ σ(Γvq)✳ ❚❤❡ ♠❛①✐♠❛❧ ✉♥✲ ❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ✐s ❞❡♥♦t❡❞ ❜② {Γu k}Uk=1✳ ❚❤❡ H, V, U ❛r❡ ❛❧❧♦✇❡❞ t♦ ❜❡ ✐♥✜♥✐t❡✳ ✶✹

✹✳✷ ❈♦♥str❛✐♥ts t♦ s❡t Γ ❇❡❢♦r❡ ✇❡ ❛♥❛❧②③❡ t❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❝❛♥❞✐❞❛t❡ s❡t ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ✇❤✐❝❤ ♠❛② ❧❡❛❞ t♦ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✱ ✇❡ ♣r❡s❡♥t s♦♠❡ r❡s✉❧ts ❢r♦♠ ❬✺❪ ✐♥ ♦✉r ♥♦t❛t✐♦♥ ❢♦r ❛ s♣❡❝✐❛❧ ❝❛s❡✳ ❚❤❡♦r❡♠ ✷ ✭❚❤❡♦r❡♠ 1✱ ▲❡♠♠❛ 1 ❬✺❪✮✳ ❈♦♥s✐❞❡r ❛ r❡❛❧ ♠❡❛s✉r❡ µ(ρ) ✇✐t❤ t❤❡ ❜♦✉♥❞❡❞ ❝♦♠♣❛❝t s✉♣♣♦rt K✳ ■❢ Z P (ρ) dµ(ρ) = 0 ✭✶✽✮ ❢♦r ❛❧❧ ♣♦❧②♥♦♠✐❛❧s P ✱ t❤❡♥ µ = 0✳ ❲❡ ✜rst s❤♦✇ t❤❛t ♦♥❧② t❤❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ❝❤♦s❡♥ ❢r♦♠ Q+ U ♠❛② ❧❡❛❞ t♦ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❛ r❛♥❞♦♠ ✇❛❧❦✳ ❚❤❡♦r❡♠ ✸✳ ■❢ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❛ r❛♥❞♦♠ ✇❛❧❦ ✐♥ t❤❡ q✉❛rt❡r✲♣❧❛♥❡ ✐s ✐♥❞✉❝❡❞ ❜② Γ ⊂ R2 +✱ t❤❡♥ Γ ⊂ Q+U✳ Pr♦♦❢✳ ❙✐♥❝❡ m s❛t✐s✜❡s ❜❛❧❛♥❝❡ ✭✸✮ ✐♥ t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ st❛t❡ s♣❛❝❡✱ ✐✳❡✳✱ (i, j)✇✐t❤ i > 0 ❛♥❞ j > 0✱ X (ρ,σ)∈Γ ρiσj_{[α(ρ, σ)(1 −} 1 X s=−1 1 X t=−1 ρ−sσ−tps,t)] = 0. ▲❡t j = 1✱ ✇❡ ✇✐❧❧ ❣❡t ❛♥ ✐♥✜♥✐t❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✇❤❡♥ i = 1, 2, · · · ✱ ∞ X p=1 (ρ(Γh_p))i−1[ X (ρ,σ)∈Γh p α(ρ, σ)(ρσ − 1 X s=−1 1 X t=−1 ρ1−sσ1−tps,t)] = 0. ✭✶✾✮ ❙✐♠✐❧❛r❧②✱ ❧❡t i = 1✱ ✇❡ ✇✐❧❧ ❣❡t ❛♥♦t❤❡r ✐♥✜♥✐t❡ s②st❡♠ ♦❢ ❡q✉❛t✐♦♥s ✇❤❡♥ j = 1, 2, · · · ✱ ∞ X q=1 (σ(Γv q))j−1[ X (ρ,σ)∈Γv q α(ρ, σ)(ρσ − 1 X s=−1 1 X t=−1 ρ1−sσ1−tps,t)] = 0. ❚❤❡ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ {α(ρk, σk)(ρkσk−P1_s=−1P1_t=−1ρ1−s_k σ1−t_k ps,t)}✱ ✇❤❡r❡ k = 1, 2, · · · ❛♥❞ (ρk, σk) ∈ Γ ✇✐❧❧ ❜❡ s❤♦✇♥ ❤❡r❡✳ ❇❡❝❛✉s❡ ♦❢ ❛ss✉♠♣✲ t✐♦♥ ✭✹✮✱ ✐✳❡✳✱ t❤❡ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ t❡r♠s ♦❢ ✇❤✐❝❤ t❤❡ s✉♠ ✐s m✱ ✇❡ ❤❛✈❡ ∞ X k=1 |α(ρk, σk)(ρkσk− 1 X s=−1 1 X t=−1 ρ1−s_k σ_k1−tps,t)| <B ∞ X k=1 |α(ρk, σk)| 1 1 − ρk 1 1 − σk <∞, ✶✺

✇❤❡r❡ B ✐s ❛ ✜♥✐t❡ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t✳ ❲❡ ❞❡✜♥❡ ❛ r❡❛❧ ♠❡❛s✉r❡ µ ❛s µ(ρ) =      P (ρ,σ)∈Γh pα(ρ, σ)(ρσ − P1 s=−1 P1 t=−1ρ1−sσ1−tps,t) ❢♦r ρ ∈ {ρ(Γh 1), ρ(Γh2), · · · }, 0 ♦t❤❡r✇✐s❡. ❲❡ ❝❛♥ ♥♦✇ ✇r✐t❡ ✭✶✾✮ ❛s ∞ X p=1 (ρ(Γh p)i−1µ(ρ(Γhp))) = Z (ρ(Γh p))i−1dµ(ρ(Γhp)) = 0, ❢♦r i = 1, 2, 3 · · · ✳ ❚❤✐s ✐♥❞✐❝❛t❡s t❤❛t Z P (ρ) dµ(ρ) = 0 ❢♦r ❛❧❧ P (ρ) = ρj _{✇❤❡r❡ j = 0, 1, 2 · · · ✳ ❍❡♥❝❡ R P (ρ) dµ(ρ) = 0 ❢♦r ❛❧❧ ♣♦❧②✲} ♥♦♠✐❛❧s✳ ▼♦r❡♦✈❡r✱ ❢♦r p = 1, 2, · · · ✱ t❤❡ s❡q✉❡♥❝❡ P(ρ,σ)∈Γh pα(ρ, σ)(ρσ − P1 s=−1 P1 t=−1ρ1−sσ1−tps,t) ✐s ❛❜s♦❧✉t❡❧② ❝♦♥✈❡r❣❡♥t✳ ❍❡♥❝❡✱ t❤❡ ❝♦♠♣❛❝t s✉♣♣♦rt ♦❢ t❤✐s s❡q✉❡♥❝❡ ✐s ❛ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳ ❚❤❡r❡❢♦r❡✱ ❜② ✉s✐♥❣ ❚❤❡✲ ♦r❡♠ ✷✱ µ = 0✱ ✐✳❡✳✱ P_(ρ,σ)∈Γh pα(ρ, σ)(ρσ − P1 s=−1 P1 t=−1ρ1−sσ1−tps,t) = 0✱ ❢♦r p = 1, 2, · · · ✳ ■♥ s✐♠✐❧❛r ❢❛s❤✐♦♥✱ ✇❡ ♦❜t❛✐♥ ❢♦r q = 1, 2, · · · ✱ t❤❛t P (ρ,σ)∈Γv qα(ρ, σ)(ρσ − P1 s=−1 P1 t=−1ρ1−sσ1−tps,t) = 0✱ ❚❤❡s❡ t✇♦ s♦❧✉t✐♦♥s ❣✉❛r❛♥t❡❡ t❤❛t ρkσk−P1_s=−1P_t=−11 ρ1−s_k σ_k1−tps,t = 0✱ ❢♦r k = 1, 2, 3 · · · ✱ ✇❤✐❝❤ ✐s t❤❡ ✐♥t❡r✐♦r ❜❛❧❛♥❝❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ♠❡❛s✉r❡ m = ρi kσ j k✱ ❤❡♥❝❡ Q(ρ, σ) = 0 ❢♦r (ρ, σ) ∈ Γ✳ ❲❡ ♥♦✇ ♠❛❦❡ t✇♦ ♦❜s❡r✈❛t✐♦♥s ♦♥ t❤❡ str✉❝t✉r❡ ♦❢ Γ ⊂ Q+ U ❢♦r ✇❤✐❝❤ t❤❡ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ❝♦♥s✐sts ♦❢ ♦♥❧② ♦♥❡ s❡t✳ ❋✐rst❧②✱ ❢♦r ❛♥② (ρ, σ) ∈ Γ t❤❡r❡ ❛❧✇❛②s ❡①✐st ❡✐t❤❡r (ρ, ˜σ) ∈ Γ✱ ✇✐t❤ ˜σ 6= σ ♦r (˜ρ, σ) ∈ Γ ✇✐t❤ ˜ ρ 6= ρ✳ ❙❡❝♦♥❞❧②✱ t❤❡ ❞❡❣r❡❡ ♦❢ Q(ρ, σ) ✐s ❛t ♠♦st t✇♦ ✐♥ ❡❛❝❤ ✈❛r✐❛❜❧❡✳ ❚❤✐s ♠❡❛♥s✱ ❢♦r ✐♥st❛♥❝❡✱ t❤❛t ✐❢ (ρ, σ) ∈ Γ ❛♥❞ (ρ, ˜σ) ∈ Γ✱ ˜σ 6= σ✱ t❤❡♥ t❤❡r❡ ❞♦❡s ♥♦t ❡①✐st (ρ, ˆσ) ∈ Γ✱ ✇❤❡r❡ ˆσ 6= σ ❛♥❞ ˆσ 6= ˜σ✳ ❇② r❡♣❡❛t✐♥❣ t❤❡ ❛❜♦✈❡ t✇♦ ❛r❣✉♠❡♥ts ❢♦r ♦t❤❡r ❡❧❡♠❡♥ts ✐♥ Γ ✐t ❢♦❧❧♦✇s t❤❛t Γ ♠✉st ❤❛✈❡ ❛ ♣❛✐r✇✐s❡✲ ❝♦✉♣❧❡❞ str✉❝t✉r❡✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ s✉❝❤ ❛ s❡t ✐s Γ = {(ρk, σk), k = 1, 2, 3 · · · }✱ ✇❤❡r❡ ρ1 = ρ2, σ1 > σ2, ρ2> ρ3, σ2= σ3, ρ3= ρ4, σ3> σ4, · · · . ✭✷✵✮ ❚❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ ❧❡❛❞s t♦ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ s❡t ✐♥ t❡r♠s ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ s❡ts ✐♥ ❛ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✻ ✭P❛✐r✇✐s❡✲❝♦✉♣❧❡❞ s❡t✮✳ ❆ s❡t Γ ⊂ Q+ U ✐s ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ Γ ❝♦♥t❛✐♥s ♦♥❧② ♦♥❡ s❡t✳ ✶✻

❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ s❤♦✇ t❤❛t ✐❢ t❤❡r❡ ❛r❡ ♠✉❧t✐♣❧❡ s❡ts ✐♥ t❤❡ ♠❛①✐♠❛❧ ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ Γ✱ t❤❡♥ t❤❡ ♠❡❛s✉r❡ ✐♥❞✉❝❡❞ ❜② t❤✐s Γ ❝❛♥♥♦t ❜❡ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡✳ ❲❡ ✜rst ✐♥tr♦❞✉❝❡ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♥♦t❛t✐♦♥✳ ❋♦r ❛♥② s❡t Γh p ❢r♦♠ t❤❡ ♠❛①✐♠❛❧ ❤♦r✐③♦♥t❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ Γ✱ ❧❡t Bh_(Γh p) = X (ρ,σ)∈Γh p α(ρ, σ)[ 1 X s=−1 ρ1−shs+ ρ1−sσps,−1
− ρ]. ❋♦r ❛♥② s❡t Γv q ❢r♦♠ t❤❡ ♠❛①✐♠❛❧ ✈❡rt✐❝❛❧❧② ✉♥❝♦✉♣❧❡❞ ♣❛rt✐t✐♦♥ ♦❢ Γ✱ ❧❡t Bv_(Γv q) = X (ρ,σ)∈Γv q α(ρ, σ)[ 1 X t=−1 σ1−tvt+ ρσ1−tp−1,t
− σ]. ❚❤❡♦r❡♠ ✹✳ ❈♦♥s✐❞❡r t❤❡ r❛♥❞♦♠ ✇❛❧❦ P ✱ ✐❢ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ m ♦❢ P ✐s m(i, j) = P_(ρ,σ)∈Γα(ρ, σ)ρi_σj_{✱ t❤❡♥ Γ ✐s ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ ❛♥❞ ❢♦r ❛❧❧} p, q ∈ {1, 2, 3, · · · }✱ Bh_(Γh p) = 0 ❛♥❞ Bv(Γvq) = 0✳ Pr♦♦❢✳ ❙✐♥❝❡ m ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ P ✱ m s❛t✐s✜❡s t❤❡ ❜❛❧❛♥❝❡ ❡q✉❛✲ t✐♦♥s ❛t st❛t❡ (i, 0) ❢♦r i = 1, 2, 3 · · · ✳ ❚❤❡r❡❢♦r❡✱ 0 = 1 X s=−1

m(i − s, 0)hs+ m(i − s, 1)ps,−1 − m(i, 0)

= X (ρ,σ)∈Γ α(ρ, σ)[ 1 X s=−1 ρi−shs+ ρi−sσps,−1
− ρi] = ∞ X p=1 ρ(Γh_p)i X (ρ,σ)∈Γh p α(ρ, σ)[ 1 X s=−1 ρ−shs+ ρ−sσps,−1
− 1] = ∞ X p=1 ρ(Γh_p)i−1Bh(Γh_p). ✭✷✶✮ ❲❡ ♥♦✇ s❤♦✇ t❤❡ ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ s❡q✉❡♥❝❡ {Bh_(Γh p)} ❢♦r p = 1, 2, 3 · · · ✳ ❇❡❝❛✉s❡ ♦❢ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t m(i, j) ✐s ❛ ✜♥✐t❡ ♠❡❛s✉r❡✱ ✇❡ ❤❛✈❡ ∞ X p=1 |Bh(Γh_p)| ≤ ∞ X k=1 α(ρk, σk)[ 1 X s=−1 ρ1−s_k hs+ ρ1−s_k σps,−1
− ρk] <∞, ✶✼

t❤❡ ❧❛st ✐♥❡q✉❛❧✐t② ❤♦❧❞s ❞✉❡ t♦ t❤❡ s❛♠❡ r❡❛s♦♥✐♥❣ t❤❛t ✐s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✸✳ ■❢ ✇❡ ❞❡✜♥❡ ❛ r❡❛❧ ♠❡❛s✉r❡ µ ❛s µ(ρ) = ( Bh_(Γh p) ✐❢ ρ = ρ(Γhp), 0 ♦t❤❡r✇✐s❡. ❲❡ ❝❛♥ ♥♦✇ ✇r✐t❡ ✭✷✶✮ ❛s ∞ X p=1 (ρ(Γh_p)i−1µ(ρ(Γh_p))) = Z (ρ(Γh_p))i−1dµ(ρ(Γh_p)) = 0, ❢♦r i = 1, 2, 3 · · · ✳ ❚❤✐s ✐♥❞✐❝❛t❡s t❤❛t Z P (ρ) dµ(ρ) = 0 ❢♦r ❛❧❧ P (ρ) = ρj _{✇❤❡r❡ j = 0, 1, 2 · · · ✳ ❍❡♥❝❡ R P (ρ) dµ(ρ) = 0 ❢♦r ❛❧❧} ♣♦❧②♥♦♠✐❛❧s✳ ▼♦r❡♦✈❡r✱ t❤❡ s❡q✉❡♥❝❡ {Bh_(Γh p)} ❢♦r p = 1, 2, · · · ✐s ❛❜s♦❧✉t❡ ❝♦♥✈❡r❣❡♥t✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦♠♣❛❝t s✉♣♣♦rt ♦❢ t❤✐s s❡q✉❡♥❝❡ ✐s ❛ ❜♦✉♥❞❡❞ ✐♥t❡r✈❛❧✳ ❍❡♥❝❡✱ ❜② ✉s✐♥❣ ❚❤❡♦r❡♠ ✷ ✇❡ ❤❛✈❡ µ = 0✱ ✇❤✐❝❤ ♠❡❛♥s Bh_(Γh p) = 0 ❢♦r p = 1, 2, · · · ✳ ❙✐♠✐❧❛r❧②✱ ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❛t Bv_(Γv q) = 0❢♦r q = 1, 2, · · · ✳ ❚❤❡s❡ t✇♦ r❡s✉❧ts ✐♠♣❧② t❤❛t Γ ✐s ❛ ♣❛✐r✇✐s❡✲❝♦✉♣❧❡❞ s❡t✳ ✹✳✸ ❈♦♥str❛✐♥t t♦ r❛♥❞♦♠ ✇❛❧❦s ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝❤❛r❛❝t❡r✐③❡ t❤❡ r❛♥❞♦♠ ✇❛❧❦s ♦❢ ✇❤✐❝❤ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♠❛② ❜❡ ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s✳ ❲❡ ✇✐❧❧ s❤♦✇ t❤❛t t❤❡ ❡①✐st❡♥❝❡ ♦❢ tr❛♥s✐t✐♦♥s t♦ ♥♦rt❤✱ ♥♦rt❤❡❛st ♦r ❡❛st ♣❧❛②s ❛♥ ❡ss❡♥t✐❛❧ r♦❧❡ ✐♥ ❞✐st✐♥❣✉✐s❤✐♥❣ s✉❝❤ ❦✐♥❞ ♦❢ r❛♥❞♦♠ ✇❛❧❦s✳ ❚❤❡ ♥❡①t r❡s✉❧t ❡①♣❧❛✐♥s t❤❛t ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛✲ s✉r❡ t❤❛t ✐s ❛♥ ✐♥✜♥✐t❡ s✉♠ ♦❢ ❣❡♦♠❡tr✐❝ t❡r♠s ❢♦r s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s✳ ❚❤❡♦r❡♠ ✺✳ ■❢ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ ❛ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ ✐s ♦❢ t❤❡ ❢♦r♠ m(i, j) = P(ρ,σ)∈Γρiσj✱ t❤❡♥ |Γ| = 1✳ Pr♦♦❢✳ ■❢ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ ❛ s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ ✐s ♦❢ t❤❡ ❢♦r♠ m(i, j) = P (ρ,σ)∈Γρiσj✱ t❤❡♥ |Γ| ≤ 2 ❜② ✉s✐♥❣ ❚❤❡♦r❡♠ ✹ ❛♥❞ ▲❡♠♠❛ ✶✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❤❛✈❡ s❤♦✇♥ ✐♥ ❬✻❪ t❤❛t t❤❡ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t✇♦ ❣❡♦✲ ♠❡tr✐❝ t❡r♠s ❝❛♥♥♦t ❜❡ t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ❢♦r ❛♥② r❛♥❞♦♠ ✇❛❧❦✱ ✇❤✐❝❤ ❝♦♠♣❧❡t❡s t❤❡ ♣r♦♦❢✳ ❲❡ ♥♦✇ ❢♦❝✉s ♦♥ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s✳ ❚❤❡ ♥❡①t t❤❡♦r❡♠ ❛♣♣❧✐❡s ▲❡♠♠❛ ✽ t♦ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦s✳ ❚❤❡♦r❡♠ ✻✳ ▲❡t t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ t❤❡ ♥♦♥✲s✐♥❣✉❧❛r r❛♥❞♦♠ ✇❛❧❦ P ❜❡ m(i, j) = P(ρ,σ)∈Γα(ρ, σ)ρiσj ✇✐t❤ |Γ| = ∞✳ ❚❤❡♥ p1,0= p1,1 = p0,1 = 0 ❛♥❞ Γ ❤❛s ❛ ✉♥✐q✉❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t ✐♥ t❤❡ ♦r✐❣✐♥✳ ✶✽

0 0.5 1 1.4 0.5 1 1.4 yt xr yb xl Q00 Q10 Q11 Q01 Q_l Qc Qr ❋✐❣✉r❡ ✼✿ ❉✐✛❡r❡♥t ♣❛rt✐t✐♦♥ ♦❢ Q+ _{❢♦r t❤❡ r❛♥❞♦♠ ✇❛❧❦ ✐♥ ❋✐❣✉r❡ ✷✭❝✮✳} Pr♦♦❢✳ ❲❡ ✇✐❧❧ ❞❡♠♦♥str❛t❡ t❤❛t ✐♥ t❤❡ ❛❜s❡♥❝❡ ♦❢ s✐♥❣✉❧❛r✐t✐❡s ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❛♥ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ m(i, j) = P(ρ,σ)∈Γα(ρ, σ)ρiσj ✇✐t❤ |Γ| = ∞✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ s❤♦✇ t❤❛t Γ ♠✉st ❤❛✈❡ ❛ s✐♥❣✉❧❛r✐t② ♦❢ Q ❛s ❛♥ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥t✳ ❚❤❡ r❡s✉❧t ♦❢ t❤❡ t❤❡♦r❡♠ t❤❡♥ ❢♦❧❧♦✇s✱ ❜② ▲❡♠♠❛ ✽✱ t❤❛t st❛t❡s t❤❡ ❛❧❣❡❜r❛✐❝ ❝✉r✈❡ Q ❤❛s ❛ s✐♥❣✉❧❛r✐t② ✐♥ ¯U ♦♥❧② ✐❢ p1,0 = p1,1 = p0,1= 0✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ✐t ✐s ✐♥ t❤❡ ♦r✐❣✐♥✳ ❙✉♣♣♦s❡ t❤❛t Q+ _{❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② s✐♥❣✉❧❛r✐t✐❡s ❛♥❞ t❤❛t m(i, j) =} P (ρ,σ)∈Γα(ρ, σ)ρiσj ✇✐t❤ |Γ| = ∞ ✐s t❤❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡ ♦❢ P ✳ ■♥ t❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣r♦♦❢ ✇❡ ✇✐❧❧ ♦❜t❛✐♥ ❛ ❝♦♥tr❛❞✐❝t✐♦♥ ❜② s❤♦✇✐♥❣ t❤❛t ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡ t❡r♠s (ρ, σ) ∈ Γ ✇✐❧❧ ❜❡ ♦✉ts✐❞❡ t❤❡ ✉♥✐t sq✉❛r❡ ❛♥❞ t❤❛t t❤❡ ♠❡❛s✉r❡ m(i, j) ❝❛♥✱ t❤❡r❡❢♦r❡✱ ♥♦t ❜❡ ✜♥✐t❡✳ ❆ss✉♠❡ xb ≤ xt✱ ❛♥♦t❤❡r ♣♦ss✐❜❧❡ ❝❛s❡ ❝❛♥ ❜❡ ❛♥❛❧②③❡❞ s✐♠✐❧❛r❧②✳ ❚♦ s✐♠♣❧✐❢② t❤❡ ♣r❡s❡♥t❛t✐♦♥ ✇❡ ✐♥tr♦❞✉❝❡ ❛❞❞✐t✐♦♥❛❧ ♥♦t❛t✐♦♥✳ ▲❡t {Ql, Qc, Qr} ❞❡♥♦t❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ Q+_{✱ ✇❤❡r❡} Ql=(x, y) ∈ Q+  x ≤ x_b , Qc =(x, y) ∈ Q+  x_b < x ≤ x_t , Qr =(x, y) ∈ Q+  x > xt . ▼♦r❡♦✈❡r✱ ❞❡♥♦t❡ t❤❡ t✇♦ ♣✐❡❝❡s ♦❢ Qc ❜② Qtc ❛♥❞ Qbc s❛t✐s❢②✐♥❣ ˜y > y ✐❢ (x, ˜_{y) ∈ Q}t c ❛♥❞ (x, y) ∈ Qbc✳ ❙✐♥❝❡ t❤❡r❡ ❛r❡ ♥♦ s✐♥❣✉❧❛r✐t✐❡s Ql, Qc ❛♥❞ Qr ❛r❡ ❛❧❧ ♥♦♥✲❡♠♣t②✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❧❡t {Γ1, . . . , ΓK} ❞❡♥♦t❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ Γ✱

✇❤❡r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ Γi ❛r❡ ❞❡♥♦t❡❞ ❛s Γi= {(ρi,1, σi,1), . . . , (ρi,L(i), σi,L(i))}

❛♥❞ ❡❛❝❤ Γi s❛t✐s✜❡s

ρi,1 > ρi,2, σi,1 = σi,2,

ρi,2 = ρi,3, σi,2 > σi,3,

ρi,3 > ρi,4, σi,3 = σi,4,

✳✳✳ ✳✳✳

ρ_i,L(i)−1 > ρ_i,L(i), σ_i,L(i)−1 = σ_i,L(i),

✭✷✷✮