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❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❉❛r✐♦ ❙♣✐r✐t♦

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ●✐✉❧✐♦ P❡r✉❣✐♥❡❧❧✐✮

❯♥✐✈❡rs✐tà ❞✐ ❘♦♠❛ ❚r❡

❆▲❛◆❚ ✺ ✕ ❏♦✐♥t ❈♦♥❢❡r❡♥❝❡s ♦♥ ❆❧❣❡❜r❛✱ ▲♦❣✐❝ ❛♥❞ ◆✉♠❜❡r ❚❤❡♦r② ✷✼ ❏✉♥❡ ✷✵✶✽

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥

❚❤❡ ❩❛r✐s❦✐ s♣❛❝❡ ❛♥❞ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②

❲❡ s❤❛❧❧ ❛❧✇❛②s ❝♦♥s✐❞❡r ❛ ❞♦♠❛✐♥ D ❛♥❞ ❛ ✜❡❧❞ K ❝♦♥t❛✐♥✐♥❣ D✳ ❩❛r(K|D) ✐s t❤❡ s❡t ♦❢ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ♦❢ K ❝♦♥t❛✐♥✐♥❣ D✳

■❢ K ✐s t❤❡ q✉♦t✐❡♥t ✜❡❧❞ ♦❢ D✱ ✇❡ s❡t ❩❛r(K|D) = ❩❛r(D)✱ ❛♥❞ ✐ts ❡❧❡♠❡♥ts ❛r❡ t❤❡ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣s ♦❢ D✳

❚❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣② ♦♥ ❩❛r(K|D) ✐s ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡ts B(x✶, . . . , xn) := {V ∈ ❩❛r(K|D) | x✶, . . . , xn ∈ V }. ❊①❝❡♣t tr✐✈✐❛❧ ❝❛s❡s✱ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣② ✐s ♥♦t T✶ ✭✐✳❡✳✱ ♥♦t ❛❧❧ ♣♦✐♥ts ❛r❡ ❝❧♦s❡❞✮❀ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✐t ✐s ♥♦t ❍❛✉s❞♦r✛✳

❚❤❡ ♦♥❧② ❝❧♦s❡❞ ♣♦✐♥ts ❛r❡ t❤❡ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥ts✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥

❙♣❡❝tr❛❧ s♣❛❝❡s

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s s♣❡❝tr❛❧ ✐❢ ✐t ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦ Spec(R) ❢♦r s♦♠❡ r✐♥❣ R✳

❙♣❡❝tr❛❧ s♣❛❝❡s ❝❛♥ ❜❡ ❝❤❛r❛❝t❡r✐③❡❞ t♦♣♦❧♦❣✐❝❛❧❧② ❬❍♦❝❤st❡r✱ ✶✾✻✾❪✳

❩❛r(K|D) ✐s ❛ s♣❡❝tr❛❧ s♣❛❝❡✿ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ✇❡ ❝❛♥ ✜♥❞ ✭❡①♣❧✐❝✐t❧②✮ ❛♥ ♦✈❡rr✐♥❣ ❑r(K|D) ♦❢ K[X] s✉❝❤ t❤❛t ❩❛r(K|D) ≃ Spec(❑r(K|D))✳ ❚❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ♦♥ ❛ s♣❡❝tr❛❧ s♣❛❝❡ X ✐s t❤❡ ❝♦❛rs❡st t♦♣♦❧♦❣② ✇❤❡r❡ t❤❡ ♦♣❡♥ ❛♥❞ ❝♦♠♣❛❝t s✉❜s❡ts ♦❢ X ❛r❡ ❜♦t❤ ♦♣❡♥ ❛♥❞ ❝❧♦s❡❞✳

X ❝♦♥s ✐s ❛ s♣❡❝tr❛❧ s♣❛❝❡ t❤❛t ✐s ❛❧s♦ ❍❛✉s❞♦r✛✳ ❚❤❡ ❝❧♦s❡❞ s❡t ♦❢ X ❝♦♥s ❛r❡ s♣❡❝tr❛❧ ✭✐♥ t❤❡ st❛rt✐♥❣ t♦♣♦❧♦❣②✮✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

■♥tr♦❞✉❝t✐♦♥

❲❤② t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②❄

❙t✉❞② ♦❢ r❡s♦❧✉t✐♦♥ ♦❢ s✐♥❣✉❧❛r✐t✐❡s ✭❩❛r✐s❦✐✮✳

❨♦✉ ♥❡❡❞ t❤❡ ❝♦♠♣❛❝t♥❡ss ♦❢ ❩❛r(K|D)✳

❙t✉❞② ♦❢ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ✈❛❧✉❛t✐♦♥ r✐♥❣s✳

■❢ X ⊆ ❩❛r(D) ✐s ❛ ❝♦♠♣❛❝t s✉❜s❡t✱ ❡❛❝❤ V ∈ X ✐s ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧✱ ❛♥❞

V ∈X mV = (✵)✱ t❤❡♥ V ∈X V ✐s ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❇é③♦✉t

❞♦♠❛✐♥ ✭✐✳❡✳✱ ❛❧❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✐❞❡❛❧s ❛r❡ ♣r✐♥❝✐♣❛❧✮ ❬❖❧❜❡r❞✐♥❣✱ ✷✵✶✼❪✳ ❙t✉❞② ♦❢ ❤♦❧♦♠♦r♣❤② ❛♥❞ r❡❛❧ ❤♦❧♦♠♦r♣❤② r✐♥❣s✳

■❢ D ✐s ❛ Prü❢❡r ❞♦♠❛✐♥✱ t❤❡♥ ❩❛r(D) ≃ Spec(D)✳

■❢ D ✐s ❛♥② ❞♦♠❛✐♥✱ t❤❡ ♠❛♣ ❩❛r(D) − → Spec(D)✱ V → mV ∩ D ✐s ❛ ❝❧♦s❡❞ ❝♦♥t✐♥✉♦✉s s✉r❥❡❝t✐♦♥✳ ■❢ D ✐s ❛♥② ❞♦♠❛✐♥✱ t❤❡ ♠❛♣ P → DP ❡♠❜❡❞s Spec(D) ✐♥ t❤❡ s❡t ♦❢ ♦✈❡rr✐♥❣s ♦❢ D✱ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss

◆♦♥✲❝♦♠♣❛❝t s✉❜s♣❛❝❡s ♦❢ ✈❛❧✉❛t✐♦♥ r✐♥❣s

❲❤✐❧❡ ❩❛r(D) ✐s ❛❧✇❛②s ❝♦♠♣❛❝t✱ t❤❡ s❛♠❡ ❞♦❡s ♥♦t ❤❛♣♣❡♥ ❢♦r s✉❜s❡ts✳ ▲❡t V ❜❡ ❛ ♠✐♥✐♠❛❧ ❡❧❡♠❡♥t ♦❢ ❩❛r(D)✳ ■❢ ❩❛r(D) \ {V } ✐s ❝♦♠♣❛❝t✱ t❤❡♥ V ✐s t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ ♦❢ D[x✶, . . . , xn]M ❢♦r s♦♠❡ x✶, . . . , xn ∈ K ❛♥❞ M ∈ Max(D[x✶, . . . , xn])✳

❚❤❡ ♣r♦♦❢ ✉s❡s t❤❡ ✐♥t❡❣r❛❧ ❝❧♦s✉r❡ ♦❢ ✐❞❡❛❧s ❛♥❞ ❛ ❝r✐t❡r✐♦♥ ❜❛s❡❞ ♦♥ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s✳

❚❤✐s ❝❛♥♥♦t ❤❛♣♣❡♥ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✿

D ✐s ◆♦❡t❤❡r✐❛♥ ❛♥❞ dim(V ) ≥ ✷❀ dim(V ) > ✷ dim(D)❀ D ✐s ❧♦❝❛❧ ❛♥❞ {P | P ∈ X} = (✵) ❢♦r s♦♠❡ ❢❛♠✐❧② X ♦❢ ♥♦♥③❡r♦ ✐♥❝♦♠♣❛r❛❜❧❡ ♣r✐♠❡ ✐❞❡❛❧s✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss

❲❤❡♥ ❩❛r(K|D) ✐s ◆♦❡t❤❡r✐❛♥

❆ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ✐s ◆♦❡t❤❡r✐❛♥ ✐❢ ❛❧❧ ✐ts s✉❜s❡ts ❛r❡ ❝♦♠♣❛❝t✳

■❢ R ✐s ❛ ◆♦❡t❤❡r✐❛♥ r✐♥❣✱ Spec(R) ✐s ❛ ◆♦❡t❤❡r✐❛♥ s♣❛❝❡✳

■❢ D = F ✐s ❛ ✜❡❧❞✱ t❤❡♥ ❩❛r(K|F) ✐s ❛ ◆♦❡t❤❡r✐❛♥ s♣❛❝❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ trdegF K ≤ ✶ ❛♥❞✱ ✐❢ X ∈ K ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F✱ t❤❡♥ ❡✈❡r② ✈❛❧✉❛t✐♦♥ ♦♥ F[X] ❡①t❡♥❞s t♦ ✜♥✐t❡❧② ♠❛♥② ✈❛❧✉❛t✐♦♥s ♦❢ K✳ ■❢ D ✐s ❧♦❝❛❧✱ t❤❡♥ ❩❛r(D) ❝❛♥ ❜❡ ◆♦❡t❤❡r✐❛♥ ♦♥❧② ✐❢ D ✐s ❛ ♣s❡✉❞♦✲✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥ ✭P❱❉✮✳

D ✐s ❛ P❱❉ ✐❢ ✐ts ♠❛①✐♠❛❧ ✐❞❡❛❧ M ✐s t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ ❛ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣✳ ■❢ ❩❛r(D) ✐s ◆♦❡t❤❡r✐❛♥✱ t❤❡♥ ❩❛r(D) \ ❩❛r♠✐♥(D) ✐s ❧✐♥❡❛r❧② ♦r❞❡r❡❞✳

❩❛r(D) ✐s ◆♦❡t❤❡r✐❛♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ Spec(D) ❛♥❞ ❩❛r(DM) ❛r❡ ◆♦❡t❤❡r✐❛♥ ❢♦r ❡✈❡r② M ∈ Max(D)✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

❈♦♠♣❛❝t♥❡ss

❖✈❡rr✐♥❣s ♦❢ ◆♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥s

▲❡t D ❜❡ ❛ ◆♦❡t❤❡r✐❛♥ r✐♥❣ ✇✐t❤ dim(D) ≥ ✷✱ ❛♥❞ q✉♦t✐❡♥t ✜❡❧❞ K✳ ❲✐t❤ t❤❡ s❛♠❡ ♠❡t❤♦❞s ❛s ❛❜♦✈❡✱ t❤❡ s♣❛❝❡ ∆ ♦❢ ◆♦❡t❤❡r✐❛♥ ✈❛❧✉❛t✐♦♥ ♦✈❡rr✐♥❣s ♦❢ D ✐s ♥♦t ❝♦♠♣❛❝t✳ ❈♦♥s✐❞❡r t❤❡ s♣❛❝❡ ❖✈❡r(D) ♦❢ ♦✈❡rr✐♥❣s ♦❢ D ✭✐✳❡✳✱ r✐♥❣s ❜❡t✇❡❡♥ D ❛♥❞ K✮ ✇✐t❤ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ s❡t ♦❢ ♦✈❡rr✐♥❣s ♦❢ D t❤❛t ❛r❡ ◆♦❡t❤❡r✐❛♥ ✐s ❝♦♠♣❛❝t ✭✐t ❤❛s ❛ ♠✐♥✐♠✉♠✮ ❜✉t ✐t ✐s ♥♦t ❛ s♣❡❝tr❛❧ s♣❛❝❡✳ ◆♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❝♦♠♣❛❝t✿

{T ∈ ❖✈❡r(D) | T ✐s ❛ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧ ❞♦♠❛✐♥}❀ {T ∈ ❖✈❡r(D) | T ✐s ❛ ❉❡❞❡❦✐♥❞ ❞♦♠❛✐♥}❀ {T ∈ ❖✈❡r(D) | T ✐s ◆♦❡t❤❡r✐❛♥ ✇✐t❤ dim(T) ≤ ✶}✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❋r♦♠ ♥♦✇ ♦♥✱ ❧❡t V ❜❡ ❛ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ✈❛❧✉❛t✐♦♥ r✐♥❣ ✇✐t❤ ♥♦♥❞✐s❝r❡t❡ ✈❛❧✉❛t✐♦♥ v✱ ✈❛❧✉❡ ❣r♦✉♣ Γv ⊆ R ❛♥❞ q✉♦t✐❡♥t ✜❡❧❞ K✳ ❲❡ ✇❛♥t t♦ st✉❞② s✉❜s❡ts ♦❢ ❩❛r(K(X)|V )✳ E := {sn}n∈N ⊂ K ✐s ❛ ♣s❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ✐❢ v(sn − sn−✶) < v(sn+✶ − sn) ❢♦r ❛❧❧ n ∈ N✱ n ≥ ✶✳ ❲❡ s❡t δn := v(sn+✶ − sn)✿ t❤❡② ❢♦r♠ ❛ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡ ✇✐t❤ ❧✐♠✐t δ ∈ R ∪ {∞} ✭❝❛❧❧❡❞ t❤❡ ❜r❡❛❞t❤ ♦❢ E✮✳ ■❢ δ = ∞✱ t❤❡♥ E ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

Ps❡✉❞♦✲❧✐♠✐ts

α ∈ K ✐s ❛ ♣s❡✉❞♦✲❧✐♠✐t ♦❢ E ✐❢ v(α − sn) < v(α − sn+✶) ❢♦r ❛❧❧ n ∈ N✳ ❨♦✉ ❝❛♥ ❛❧s♦ ❝♦♥s✐❞❡r ♣s❡✉❞♦✲❧✐♠✐ts ✐♥ K ✇✐t❤ r❡s♣❡❝t t♦ ❛♥ ❡①t❡♥s✐♦♥ u ♦❢ v✳ ❋✐① ❛♥ ❡①t❡♥s✐♦♥ u✳ ❚❤❡ s❡t ♦❢ ♣s❡✉❞♦✲❧✐♠✐ts ♦❢ E ✐♥ K✱ ✐❢ ♥♦♥❡♠♣t②✱ ✐s ❛ ❜❛❧❧ ♦❢ r❛❞✐✉s e−δ✳ ❲❡ ❞✐st✐♥❣✉✐s❤ t✇♦ t②♣❡s ♦❢ ♣s❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✿

E ✐s ♦❢ ❛❧❣❡❜r❛✐❝ ✐❢ v(f (sn)) ✐s ❞❡✜♥✐t✐✈❡❧② ✐♥❝r❡❛s✐♥❣ ❢♦r s♦♠❡ ♣♦❧②♥♦♠✐❛❧ f ∈ K[X]❀ E ✐s ♦❢ tr❛♥s❝❡♥❞❡♥t❛❧ ✐❢ v(f (sn)) ✐s ❞❡✜♥✐t✐✈❡❧② ❝♦♥st❛♥t ❢♦r ❛❧❧ f ∈ K[X]✳

❊q✉✐✈❛❧❡♥t❧②✱ ✐t ✐s ❛❧❣❡❜r❛✐❝ ✐❢ E ❤❛s ♣s❡✉❞♦✲❧✐♠✐ts ✐♥ K✱ tr❛♥s❝❡♥❞❡♥t❛❧ ✐❢ ✐t ❞♦❡s♥✬t✳ ■❢ E ✐s ❛ ❈❛✉❝❤② s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ♣s❡✉❞♦✲❧✐♠✐t ✐♥ K✱ ✇❤✐❝❤ ✐s ❛❧❣❡❜r❛✐❝ ♦✈❡r K ✐❢ E ✐s ❛❧❣❡❜r❛✐❝ ❛♥❞ tr❛♥s❝❡♥❞❡♥t❛❧ ✐❢ E ✐s tr❛♥s❝❡♥❞❡♥t❛❧✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❱❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ❛ss♦❝✐❛t❡❞ t♦ E ✭✶✮

❲❡ ✇❛♥t t♦ ❛ss♦❝✐❛t❡ t♦ ❛ ♣s❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ V t♦ K(X)✳ ▲❡t E := {sn}n∈N✳ ❋♦r ❡✈❡r② φ ∈ K(X)✱ ❧❡t wE(φ) := lim

n→∞ v(φ(sn))

■❢ E ✐s tr❛s❝❡♥❞❡♥t✱ wE ✐s ❛ ✈❛❧✉❛t✐♦♥✳ ■❢ E ✐s ❛❧❣❡❜r❛✐❝ ❛♥❞ δE < ∞✱ wE ✐s ❛ ✈❛❧✉❛t✐♦♥✳ ■❢ E ✐s ❛❧❣❡❜r❛✐❝ ❛♥❞ ❈❛✉❝❤②✱ wE ✐s ❛ ♣s❡✉❞♦✲✈❛❧✉❛t✐♦♥ ♦♥ K[X]✳

■❢ wE ✐s ❛ ✈❛❧✉❛t✐♦♥✱ ✇❡ ❞❡♥♦t❡ ❜② WE ✐ts ✈❛❧✉❛t✐♦♥ r✐♥❣✳ WE ✐s ❛❧✇❛②s ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❛♥❞ ❛♥ ❡①t❡♥s✐♦♥ ♦❢ V ✳ ■❢ K ✐s ❛❧❣❡❜r❛✐❝❛❧❧② ❝❧♦s❡❞✱ ❡✈❡r② r❛♥❦✲♦♥❡ ❡①t❡♥s✐♦♥ ♦❢ V t♦ K(X) ✐s ✐♥ t❤❡ ❢♦r♠ WE ❬❖str♦✇s❦✐✱ ✶✾✸✺❪✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❱❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ❛ss♦❝✐❛t❡❞ t♦ E ✭✷✮

■❢ E = {sn}n∈N✱ ✇❡ ❞❡✜♥❡ VE := {φ ∈ K(X) | φ(sn) ∈ V ❢♦r ❛❧❧ ❧❛r❣❡ n}.

VE ✇❛s ❞❡✜♥❡❞ ✐♥ ❬▲♦♣❡r ❛♥❞ ❲❡r♥❡r✱ ✷✵✶✻❪ ❢♦r E tr❛♥s❝❡♥❞❡♥t❛❧ ♦r ✇❤❡♥ δE = ∞✳

VE ✐s ❛❧✇❛②s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ V t♦ K(X)✳ ■❢ E ✐s ♦❢ tr❛♥s❝❡♥❞❡♥t❛❧ t②♣❡✱ VE = WE ❤❛s r❛♥❦ ✶✳ ■❢ E ✐s ♦❢ ❛❧❣❡❜r❛✐❝ t②♣❡✱ t❤❡♥✿

✐❢ δ ✐s t♦rs✐♦♥ ♦✈❡r Γv✱ t❤❡♥ VE ❤❛s r❛♥❦ ✷ ❛♥❞ WE ❤❛s r❛♥❦ ✶❀ ✐❢ δ ✐s ♥♦t t♦rs✐♦♥ ♦✈❡r Γv✱ t❤❡♥ VE = WE ❤❛s r❛♥❦ ✶❀ ✐❢ δ = ∞✱ t❤❡♥ VE ❤❛s r❛♥❦ ✷✱ ❛♥❞ ✐ts ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ♦✈❡rr✐♥❣ ✐s K[X](q)✱ ✇❤❡r❡ q ✐s t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❧✐♠✐t ♦❢ E✳

❲❡ ❝❛♥ ❞❡s❝r✐❜❡ ❡①♣❧✐❝✐t❧② t❤❡ ✈❛❧✉❛t✐♦♥ vE✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❊q✉✐✈❛❧❡♥❝❡

❲❡ s❛② t❤❛t E ❛♥❞ F ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ δE = δF ❛♥❞✱ ❢♦r ❡✈❡r② k ∈ N✱ t❤❡r❡ ❛r❡ i✵, j✵ ∈ N s✉❝❤ t❤❛t✱ ✇❤❡♥❡✈❡r i ≥ i✵✱ j ≥ j✵ t❤❡♥ v(si − tj) > v(tk+✶ − tk).

❚❤✐s ✐s ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❝♦♥❝❡♣t ♦❢ ❡q✉✐✈❛❧❡♥❝❡ ❜❡t✇❡❡♥ ❈❛✉❝❤② s❡q✉❡♥❝❡s✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿

E ❛♥❞ F ❛r❡ ❡q✉✐✈❛❧❡♥t❀ VE = VF❀ WE = WF ✭✇❤❡♥ t❤❡② ❛r❡ ❞❡✜♥❡❞✮✳

■❢ E ❛♥❞ F ❛r❡ ❡q✉✐✈❛❧❡♥t✱ t❤❡♥ t❤❡② ❛r❡ ❡✐t❤❡r ❜♦t❤ ❛❧❣❡❜r❛✐❝ ♦r ❜♦t❤ tr❛♥s❝❡♥❞❡♥t❛❧✳ ■❢ E ❛♥❞ F ❛r❡ ❛❧❣❡❜r❛✐❝✱ t❤❡♥ t❤❡② ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♣s❡✉❞♦✲❧✐♠✐ts ✐♥ K✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❚❤❡ s♣❛❝❡ W

▲❡t W := {WE | E ✐s ❛ ♥♦t ❛❧❣❡❜r❛✐❝ ❛♥❞ ❈❛✉❝❤②} ⊆ ❩❛r(K(X)|V ). ❚❤❡ ❩❛r✐s❦✐ ❛♥❞ t❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ❛❣r❡❡ ♦♥ W✳ W ✐s r❡❣✉❧❛r ✭✐✳❡✳✱ ❛ ♣♦✐♥t ❛♥❞ ❛ ❝❧♦s❡❞ s❡t ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ❜② ♦♣❡♥ s❡ts✮✳ W ✐s ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧ ✭✐✳❡✳✱ ❛ ♣♦✐♥t ❛♥❞ ❛ ❝❧♦s❡❞ s❡t ❝❛♥ ❜❡ s❡♣❛r❛t❡❞ ❜② ❝❧♦♣❡♥ s❡ts✮✳ W ✐s ♥♦t ❝♦♠♣❛❝t✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❚❤❡ s♣❛❝❡ V

▲❡t V := {VE | E ✐s ❛ ♣s❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡} ⊆ ❩❛r(K(X)|V ). ❊❛❝❤ ♣♦✐♥t ♦❢ V ✐s ❝❧♦s❡❞✳ V ✐s r❡❣✉❧❛r ✐♥ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ❩❛r✐s❦✐ ❛♥❞ t❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ❛❣r❡❡ ♦♥ V ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ r❡s✐❞✉❡ ✜❡❧❞ ♦❢ V ✐s ✜♥✐t❡✳ ❊♥❞♦✇ V ✇✐t❤ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ♠❛♣ W − → V WE − → VE ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ ✐♥❥❡❝t✐✈❡✱ ❜✉t ♥♦t ❛ t♦♣♦❧♦❣✐❝❛❧ ❡♠❜❡❞❞✐♥❣✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❋✐①❡❞ ❜r❡❛❞t❤ ✭✶✮

▲❡t V(•, δ) := {VE ∈ V | δE = δ}. ■❢ E := {sn}n∈N✱ F := {tn}n∈N✱ s❡t dδ(VE, VF) := lim

n→∞ max{d(sn, tn) − e−δ, ✵}.

dδ ✐s ❛♥ ✉❧tr❛♠❡tr✐❝ ❞✐st❛♥❝❡ ♦❢ V(•, δ)✳ ■❢ E, F ∈ V(•, δ) ❛r❡ ❛❧❣❡❜r❛✐❝✱ VE = VF✱ ❛♥❞ αE, αF ❛r❡ ♣s❡✉❞♦✲❧✐♠✐ts✱ t❤❡♥ dδ(VE, VF) = d(αE, αF) − e−δ = e−v(αE −αF ) − e−δ V(•, ∞) ✐s ❡ss❡♥t✐❛❧❧② K✱ ❛♥❞ d∞ r❡❞✉❝❡s t♦ t❤❡ ❞✐st❛♥❝❡ ✐♥❞✉❝❡❞ ❜② v✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❋✐①❡❞ ❜r❡❛❞t❤ ✭✷✮

V(•, δ) ✐s ❝♦♠♣❧❡t❡ ✉♥❞❡r dδ✳ ❆♥ ✐♠♣♦rt❛♥t ❞❡♥s❡ s✉❜s❡t ✐s VK(•, δ) := {VE ∈ V(•, δ) | E ❤❛s ❛ ♣s❡✉❞♦✲❧✐♠✐t ✐♥ K}.

VK(•, ∞) ❝♦rr❡s♣♦♥❞ t♦ K✳

❚❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✱ t❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ❛♥❞ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❜② dδ ❝♦✐♥❝✐❞❡ ♦♥ V(•, δ)✳

■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ t✇♦ t♦♣♦❧♦❣✐❡s ❛❣r❡❡ ♦♥ ❡❛❝❤ s❧✐❝❡✱ ❜✉t ❞♦ ♥♦t ❛❣r❡❡ ✭✐♥ ❣❡♥❡r❛❧✮ ♦♥ t❤❡ ✇❤♦❧❡ V✳

❚❤❡ ✈❛r✐♦✉s dδ ❝❛♥♥♦t ❜❡ ✉♥✐✜❡❞ t♦ ❛ ♠❡tr✐❝ ♦♥ t❤❡ ✇❤♦❧❡ V✳

V(•, δ) ✐s ♥♦t ❝❧♦s❡❞✦

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❋✐①❡❞ ❝❡♥t❡r

❚❛❦❡ β ∈ K✱ ❛♥❞ ❛ ✜①❡❞ ❡①t❡♥s✐♦♥ u ♦❢ v t♦ K✳ ▲❡t Vu(β, •) := {VE ∈ V | β ✐s ❛ ♣s❡✉❞♦✲❧✐♠✐t ♦❢ E ✇rt u}. Vu(β, •) ✐s ❝❧♦s❡❞ ✐♥ V✳ ❚❤❡ ❩❛r✐s❦✐ ❛♥❞ t❤❡ ❝♦♥str✉❝t✐❜❧❡ t♦♣♦❧♦❣② ❛❣r❡❡ ♦♥ Vu(β, •)✳ ❚✇♦ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ✐♥ Vu(β, •) ❛r❡ ❞✐st✐♥❣✉✐s❤❡❞ ❜② t❤❡ ❜r❡❛❞t❤❀ ❤❡♥❝❡✱ ✇❡ ✇❛♥t t♦ ❝♦♥s✐❞❡r t❤❡ ♠❛♣ Vu(β, •) − → (−∞, +∞] VE − → δE ■ts r❛♥❣❡ ✐s (−∞, δ(β, K)]✱ ✇❤❡r❡ δ(β, K) := sup{u(β − x) | x ∈ K} ❞❡♣❡♥❞s ♦♥ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ β ❛♥❞ K✳ ❲❤❛t ❛❜♦✉t t❤❡ t♦♣♦❧♦❣✐❝❛❧ ❧❡✈❡❧❄

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❋✐①❡❞ ❝❡♥t❡r ✭✷✮

❖♥ (−∞, δ(β, K)]✱ ✇❡ ♣✉t t❤❡ t♦♣♦❧♦❣② ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡ts (a, b]✱ ✇✐t❤ b ∈ QΓv✳

❚❤✐s ✐s ❛ ✈❛r✐❛♥t ♦❢ t❤❡ ✉♣♣❡r ❧✐♠✐t t♦♣♦❧♦❣②✳

❯♥❞❡r t❤✐s t♦♣♦❧♦❣②✱ t❤❡ ♠❛♣ VE → δE ❜❡❝♦♠❡s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❡q✉✐✈❛❧❡♥t✿

Vu(β, •) ✐s ♠❡tr✐③❛❜❧❡❀ t❤❡r❡ ✐s ❛♥ ✉❧tr❛♠❡tr✐❝ ❞✐st❛♥❝❡ ♦♥ Vu(β, •)❀ Vu(β, •) ✐s s❡❝♦♥❞✲❝♦✉♥t❛❜❧❡❀ Γv ✐s ❝♦✉♥t❛❜❧❡✳

■❢ Γv ✐s ♥♦t ❝♦✉♥t❛❜❧❡✱ t❤❡♥ ❩❛r(K(X)|V )❝♦♥s ✐s ♥♦t ♠❡tr✐③❛❜❧❡✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❖♣❡♥ ♣r♦❜❧❡♠s

■s V ③❡r♦✲❞✐♠❡♥s✐♦♥❛❧❄ ■❢ Γv ✐s ❝♦✉♥t❛❜❧❡✱ ❛r❡ V ❛♥❞ W ♠❡tr✐③❛❜❧❡❄ ■❢ Γv ✐s ❝♦✉♥t❛❜❧❡✱ ✐s ❩❛r(K(X)|V )❝♦♥s ♠❡tr✐③❛❜❧❡❄ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✇❤❡♥ ✐s ❩❛r(K|D)❝♦♥s ♠❡tr✐③❛❜❧❡❄ ■❢ ❛♥② ♦❢ t❤❡♠ ✐s ♠❡tr✐③❛❜❧❡✱ ❝❛♥ ✇❡ ✜♥❞ ❛♥ ✉❧tr❛♠❡tr✐❝ ❞✐st❛♥❝❡❄

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❇✐❜❧✐♦❣r❛♣❤②

▲♦♣❡r✱ ❆✳ ❛♥❞ ❲❡r♥❡r✱ ◆✳✱ Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s ❛♥❞ Pr✉❢❡r ❞♦♠❛✐♥s ♦❢ ✐♥t❡❣❡r✲✈❛❧✉❡❞ ♣♦❧②♥♦♠✐❛❧s✱ ❏✳ ❈♦♠♠✉t✳ ❆❧❣❡❜r❛ ✽ ✭✷✵✶✻✮✱ ♥♦✳ ✸✱ ✹✶✶✲✹✷✾✳ ❍♦❝❤st❡r✱ ▼✳✱ Pr✐♠❡ ✐❞❡❛❧ str✉❝t✉r❡ ✐♥ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣s✳ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✭✶✾✻✾✮✱ ✶✹✷✿✹✸✕✻✵✳ ❖❧❜❡r❞✐♥❣✱ ❇✳✱ ❆ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧ t❤❡♦r❡♠ ❢♦r ❝♦♠♣❛❝t s❡ts ♦❢ r❛♥❦ ♦♥❡ ✈❛❧✉❛t✐♦♥ r✐♥❣s✳ ❏✳ ❆❧❣❡❜r❛ ✹✽✾ ✭✷✵✶✼✮✱ ✸✾✾✕✹✷✻✳ ❖str♦✇s❦✐✱ ❆✳✱ ❯♥t❡rs✉❝❤✉♥❣❡♥ ③✉r ❛r✐t❤♠❡t✐s❝❤❡♥ ❚❤❡♦r✐❡ ❞❡r ❑ör♣❡r✱ ▼❛t❤✳ ❩✳ ✸✾ ✭✶✾✸✺✮✱ ✷✻✾✕✹✵✹✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡

Ps❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s

❇✐❜❧✐♦❣r❛♣❤② ✭✷✮

P❡r✉❣✐♥❡❧❧✐✱ ●✳ ❛♥❞ ❙♣✐r✐t♦✱ ❉✳✱ ❚❤❡ ❩❛r✐s❦✐✲❘✐❡♠❛♥♥ s♣❛❝❡ ♦❢ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥s ❛ss♦❝✐❛t❡❞ t♦ ♣s❡✉❞♦✲❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡s✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳ ❙♣✐r✐t♦✱ ❉✳✱ ◆♦♥✲❝♦♠♣❛❝t s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡ ♦❢ ❛♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥✳ ■❧❧✐♥♦✐s ❏✳ ▼❛t❤✳ ✻✵✭✸✲✹✮ ✭✷✵✶✼✮✱ ✼✾✶✕✽✵✾✳ ❙♣✐r✐t♦✱ ❉✳✱ ❲❤❡♥ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡ ✐s ❛ ◆♦❡t❤❡r✐❛♥ s♣❛❝❡✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✳

❉❛r✐♦ ❙♣✐r✐t♦ ❚♦♣♦❧♦❣✐❝❛❧ ♣r♦♣❡rt✐❡s ♦❢ s✉❜s❡ts ♦❢ t❤❡ ❩❛r✐s❦✐ s♣❛❝❡