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■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠

❆ ❞❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❝♦♠♠✉t❛t✐✈❡ r❛♥❦ ♦❢ ♠❛tr✐① s♣❛❝❡s

▼❛r❦✉s ❇❧äs❡r1✱ ●♦r❛✈ ❏✐♥❞❛❧2 ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡②2

1❙❛❛r❧❛♥❞ ❯♥✐✈❡rs✐t② 2▼❛①✲P❧❛♥❝❦✲■♥st✐t✉t❡ ❢♦r ■♥❢♦r♠❛t✐❝s

❈❈❈ ✷✵✶✼ ✵✾✴✵✼✴✷✵✶✼

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠

✶

■♥tr♦❞✉❝t✐♦♥ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

✷

▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙❡t✉♣

F ❜❡ ❛♥② ✜❡❧❞✱ n ∈ Z>0✳

Fn×n ✐s t❤❡ ✭✈❡❝t♦r✮ s♣❛❝❡ ♦❢ ❛❧❧ n × n ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥ F✳

❋♦r ✈❡❝t♦r s♣❛❝❡s V, W

❯s❡ ♥♦t❛t✐♦♥ V ≤ W t♦ ❞❡♥♦t❡ t❤❛t V ✐s ❛ s✉❜s♣❛❝❡ ♦❢ W✳

❉❡✜♥✐t✐♦♥ ✭▼❛tr✐① s♣❛❝❡✮ ❆ ✈❡❝t♦r s♣❛❝❡ B ≤ Fn×n ✐s ❝❛❧❧❡❞ ❛ ♠❛tr✐① s♣❛❝❡✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙❡t✉♣

F ❜❡ ❛♥② ✜❡❧❞✱ n ∈ Z>0✳

Fn×n ✐s t❤❡ ✭✈❡❝t♦r✮ s♣❛❝❡ ♦❢ ❛❧❧ n × n ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥ F✳

❋♦r ✈❡❝t♦r s♣❛❝❡s V, W

❯s❡ ♥♦t❛t✐♦♥ V ≤ W t♦ ❞❡♥♦t❡ t❤❛t V ✐s ❛ s✉❜s♣❛❝❡ ♦❢ W✳

❉❡✜♥✐t✐♦♥ ✭▼❛tr✐① s♣❛❝❡✮ ❆ ✈❡❝t♦r s♣❛❝❡ B ≤ Fn×n ✐s ❝❛❧❧❡❞ ❛ ♠❛tr✐① s♣❛❝❡✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙❡t✉♣

F ❜❡ ❛♥② ✜❡❧❞✱ n ∈ Z>0✳

Fn×n ✐s t❤❡ ✭✈❡❝t♦r✮ s♣❛❝❡ ♦❢ ❛❧❧ n × n ♠❛tr✐❝❡s ✇✐t❤ ❡♥tr✐❡s ✐♥ F✳

❋♦r ✈❡❝t♦r s♣❛❝❡s V, W

❯s❡ ♥♦t❛t✐♦♥ V ≤ W t♦ ❞❡♥♦t❡ t❤❛t V ✐s ❛ s✉❜s♣❛❝❡ ♦❢ W✳

❉❡✜♥✐t✐♦♥ ✭▼❛tr✐① s♣❛❝❡✮ ❆ ✈❡❝t♦r s♣❛❝❡ B ≤ Fn×n ✐s ❝❛❧❧❡❞ ❛ ♠❛tr✐① s♣❛❝❡✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

Pr♦❜❧❡♠

Pr♦❜❧❡♠

• ✐✈❡♥ ❛ ♠❛tr✐① s♣❛❝❡ B ≤ Fn×n ❛s ✐♥♣✉t✱ ❝♦♠♣✉t❡ ✐ts ✏r❛♥❦✑✳ B ✐s

❣✐✈❡♥ ❛s ✐♥♣✉t ❜② ✐ts s❡t ♦❢ ❣❡♥❡r❛t♦rs✱ ✐✳❡✱ B = B1, B2, . . . , Bm. ❚✇♦ ♥♦t✐♦♥s ♦❢ r❛♥❦✳

❈♦♠♠✉t❛t✐✈❡ r❛♥❦✳ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

Pr♦❜❧❡♠

Pr♦❜❧❡♠

• ✐✈❡♥ ❛ ♠❛tr✐① s♣❛❝❡ B ≤ Fn×n ❛s ✐♥♣✉t✱ ❝♦♠♣✉t❡ ✐ts ✏r❛♥❦✑✳ B ✐s

❣✐✈❡♥ ❛s ✐♥♣✉t ❜② ✐ts s❡t ♦❢ ❣❡♥❡r❛t♦rs✱ ✐✳❡✱ B = B1, B2, . . . , Bm. ❚✇♦ ♥♦t✐♦♥s ♦❢ r❛♥❦✳

❈♦♠♠✉t❛t✐✈❡ r❛♥❦✳ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❈♦♠♠✉t❛t✐✈❡ r❛♥❦

❉❡✜♥✐t✐♦♥ ✭❈♦♠♠✉t❛t✐✈❡ r❛♥❦✮ B ≤ Fn×n ❛♥② ♠❛tr✐① s♣❛❝❡✱ t❤❡♥ ❈♦♠♠✉t❛✐✈❡ r❛♥❦ ♦❢ B = rank(B) = max{rank(B) | B ∈ B}✳ B ≤ Fn×n ✐s ❝❛❧❧❡❞ ❢✉❧❧✲r❛♥❦ ✐❢ rank(B) = n.

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❈♦♠♠✉t❛t✐✈❡ r❛♥❦

❉❡✜♥✐t✐♦♥ ✭❈♦♠♠✉t❛t✐✈❡ r❛♥❦✮ B ≤ Fn×n ❛♥② ♠❛tr✐① s♣❛❝❡✱ t❤❡♥ ❈♦♠♠✉t❛✐✈❡ r❛♥❦ ♦❢ B = rank(B) = max{rank(B) | B ∈ B}✳ B ≤ Fn×n ✐s ❝❛❧❧❡❞ ❢✉❧❧✲r❛♥❦ ✐❢ rank(B) = n.

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆ ❞✐✛❡r❡♥t ❋♦r♠✉❧❛t✐♦♥

▼❛tr✐① s♣❛❝❡ B = B1, B2, . . . , Bm ≤ Fn×n✱ ❝♦♥s✐❞❡r t❤❡ ♠❛tr✐①

B = x1B1 + x2B2 + . . . + xmBm ♦✈❡r t❤❡ ✜❡❧❞ F(x1, x2, . . . , xm) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳

❋❛❝t ■❢ |F| > n t❤❡♥ rank(B) = rank(B)✳

• ✐✈❡s ❛ r❛♥❞♦♠✐③❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ✉s✐♥❣

❙❝❤✇❛rt③✕❩✐♣♣❡❧ ❧❡♠♠❛✳

❊✈❡♥ ❛♥ ❘◆❈ ❛❧❣♦r✐t❤♠✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆ ❞✐✛❡r❡♥t ❋♦r♠✉❧❛t✐♦♥

▼❛tr✐① s♣❛❝❡ B = B1, B2, . . . , Bm ≤ Fn×n✱ ❝♦♥s✐❞❡r t❤❡ ♠❛tr✐①

B = x1B1 + x2B2 + . . . + xmBm ♦✈❡r t❤❡ ✜❡❧❞ F(x1, x2, . . . , xm) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳

❋❛❝t ■❢ |F| > n t❤❡♥ rank(B) = rank(B)✳

• ✐✈❡s ❛ r❛♥❞♦♠✐③❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ✉s✐♥❣

❙❝❤✇❛rt③✕❩✐♣♣❡❧ ❧❡♠♠❛✳

❊✈❡♥ ❛♥ ❘◆❈ ❛❧❣♦r✐t❤♠✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆ ❞✐✛❡r❡♥t ❋♦r♠✉❧❛t✐♦♥

▼❛tr✐① s♣❛❝❡ B = B1, B2, . . . , Bm ≤ Fn×n✱ ❝♦♥s✐❞❡r t❤❡ ♠❛tr✐①

B = x1B1 + x2B2 + . . . + xmBm ♦✈❡r t❤❡ ✜❡❧❞ F(x1, x2, . . . , xm) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s✳

❋❛❝t ■❢ |F| > n t❤❡♥ rank(B) = rank(B)✳

• ✐✈❡s ❛ r❛♥❞♦♠✐③❡❞ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠ ✉s✐♥❣

❙❝❤✇❛rt③✕❩✐♣♣❡❧ ❧❡♠♠❛✳

❊✈❡♥ ❛♥ ❘◆❈ ❛❧❣♦r✐t❤♠✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❖✉r ❝♦♥tr✐❜✉t✐♦♥

❆ ❞❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❈♦♠♠✉t❛t✐✈❡ r❛♥❦✳ ❚❤❡♦r❡♠ ❋♦r ❛♥② ▼❛tr✐① s♣❛❝❡ B ≤ Fn×n ❛s ✐♥♣✉t✱ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ♦✉t♣✉ts ❛ ♠❛tr✐① A ∈ B s✉❝❤ t❤❛t rank(A) ≥ (1 − ǫ) rank(B). ❆❧❣♦r✐t❤♠ r✉♥s ✐♥ t✐♠❡ nO( 1

ǫ)✳ ▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦

❉❡✜♥✐t✐♦♥ ✭c✲s❤r✉♥❦ s✉❜s♣❛❝❡✮ V ≤ Fn ✐s ❛ c✲s❤r✉♥❦ s✉❜s♣❛❝❡ ♦❢ B ≤ Fn×n ✱ ✐❢ rank(BV) ≤ dim(V) − c✳ ❉❡✜♥✐t✐♦♥ ✭◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦✮ B ≤ Fn×n ❛♥② ♠❛tr✐① s♣❛❝❡✱ ✐❢ r = max{c | ∃ c✲s❤r✉♥❦ s✉❜s♣❛❝❡♦❢ B} t❤❡♥ ◆♦♥✲❝♦♠♠✉t❛✐✈❡ r❛♥❦ ♦❢ B = ncr(B) = n − r✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦

❉❡✜♥✐t✐♦♥ ✭c✲s❤r✉♥❦ s✉❜s♣❛❝❡✮ V ≤ Fn ✐s ❛ c✲s❤r✉♥❦ s✉❜s♣❛❝❡ ♦❢ B ≤ Fn×n ✱ ✐❢ rank(BV) ≤ dim(V) − c✳ ❉❡✜♥✐t✐♦♥ ✭◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦✮ B ≤ Fn×n ❛♥② ♠❛tr✐① s♣❛❝❡✱ ✐❢ r = max{c | ∃ c✲s❤r✉♥❦ s✉❜s♣❛❝❡♦❢ B} t❤❡♥ ◆♦♥✲❝♦♠♠✉t❛✐✈❡ r❛♥❦ ♦❢ B = ncr(B) = n − r✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

Pr♦❜❧❡♠

▲❡♠♠❛ ✭❋♦rt✐♥ ❛♥❞ ❘❡✉t❡♥❛✉❡r✱ ✷✵✵✹✮ rank(B) ≤ ncr(B) ≤ 2 · rank(B) ▲❡♠♠❛ ✭❉❡r❦s❡♥ ❛♥❞ ▼❛❦❛♠✱ ✷✵✶✻✮ ❚❤❡r❡ ❡①✐st B ≤ Fn×n s✉❝❤ t❤❛t

ncr(B) rank(B) ❣❡ts ❛r❜✐tr❛r✐❧② ❝❧♦s❡ t♦ 2

❛s n → ∞✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

Pr♦❜❧❡♠

▲❡♠♠❛ ✭❋♦rt✐♥ ❛♥❞ ❘❡✉t❡♥❛✉❡r✱ ✷✵✵✹✮ rank(B) ≤ ncr(B) ≤ 2 · rank(B) ▲❡♠♠❛ ✭❉❡r❦s❡♥ ❛♥❞ ▼❛❦❛♠✱ ✷✵✶✻✮ ❚❤❡r❡ ❡①✐st B ≤ Fn×n s✉❝❤ t❤❛t

ncr(B) rank(B) ❣❡ts ❛r❜✐tr❛r✐❧② ❝❧♦s❡ t♦ 2

❛s n → ∞✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❲❤② st✉❞② t❤✐s ♣r♦❜❧❡♠❄

• ❡♥❡r❛❧✐③❡s s❡✈❡r❛❧ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s ❢r♦♠ ❛❧❣❡❜r❛ ❛♥❞

❝♦♠❜✐♥❛t♦r✐❝s✳

❇✐♣❛rt✐t❡ ♠❛t❝❤✐♥❣ ▲✐♥❡❛r ▼❛tr♦✐❞ ✐♥t❡rs❡❝t✐♦♥✳ ▼❛①✐♠✉♠ ♠❛t❝❤✐♥❣ ▲✐♥❡❛r ♠❛tr♦✐❞ ♣❛r✐t② ♣r♦❜❧❡♠

P♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t② t❡st✐♥❣✭P■❚✮ ♦❢ ❆❧❣❡❜r❛✐❝ ❜r❛♥❝❤✐♥❣ ♣r♦❣r❛♠s✭❆❇P✮

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙♣❡❝✐❛❧ ❝❛s❡s

◆P✲❝♦♠♣❧❡t❡ ✇❤❡♥ t❤❡ ✜❡❧❞ F ✐s ♦❢ ❝♦♥st❛♥t s✐③❡✳ ❉❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✇❤❡♥ Bi✬s ❛❧❧ ❛r❡ ♦❢ r❛♥❦ ✶✳

❙✉❜s✉♠❡s ❜✐♣❛rt✐t❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣✱ ❧✐♥❡❛r ♠❛tr♦✐❞ ✐♥t❡rs❡❝t✐♦♥✳ ❊✈❡♥ ❛ q✉❛s✐✲◆❈ ❛❧❣♦r✐t❤♠ ❜② ❬●✉r❥❛r ❛♥❞ ❚❤✐❡r❛✉❢✱ ✷✵✶✻❪✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙♣❡❝✐❛❧ ❝❛s❡s

◆P✲❝♦♠♣❧❡t❡ ✇❤❡♥ t❤❡ ✜❡❧❞ F ✐s ♦❢ ❝♦♥st❛♥t s✐③❡✳ ❉❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✇❤❡♥ Bi✬s ❛❧❧ ❛r❡ ♦❢ r❛♥❦ ✶✳

❙✉❜s✉♠❡s ❜✐♣❛rt✐t❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣✱ ❧✐♥❡❛r ♠❛tr♦✐❞ ✐♥t❡rs❡❝t✐♦♥✳ ❊✈❡♥ ❛ q✉❛s✐✲◆❈ ❛❧❣♦r✐t❤♠ ❜② ❬●✉r❥❛r ❛♥❞ ❚❤✐❡r❛✉❢✱ ✷✵✶✻❪✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❙♣❡❝✐❛❧ ❝❛s❡s

◆P✲❝♦♠♣❧❡t❡ ✇❤❡♥ t❤❡ ✜❡❧❞ F ✐s ♦❢ ❝♦♥st❛♥t s✐③❡✳ ❉❡t❡r♠✐♥✐st✐❝ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✇❤❡♥ Bi✬s ❛❧❧ ❛r❡ ♦❢ r❛♥❦ ✶✳

❙✉❜s✉♠❡s ❜✐♣❛rt✐t❡ ♠❛①✐♠✉♠ ♠❛t❝❤✐♥❣✱ ❧✐♥❡❛r ♠❛tr♦✐❞ ✐♥t❡rs❡❝t✐♦♥✳ ❊✈❡♥ ❛ q✉❛s✐✲◆❈ ❛❧❣♦r✐t❤♠ ❜② ❬●✉r❥❛r ❛♥❞ ❚❤✐❡r❛✉❢✱ ✷✵✶✻❪✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆❧❣♦r✐t❤♠s ❢♦r ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦

• ✉r✈✐ts✱ ✷✵✵✹ ✿ ❉❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠s ❢♦r

✏❝♦♠♣r❡ss✐♦♥ s♣❛❝❡s✑

▼❛tr✐① s♣❛❝❡ B ✐s ❛ ❝♦♠♣r❡ss✐♦♥ s♣❛❝❡ ✐❢ rank(B) = ncr(B)✳

❚❤❡♦r❡♠ ✭●●❖❲ ✷✵✶✺✱ ■✈❛♥②♦s ❡t ❛❧✳✱✷✵✶✺ ✮ ❚❤❡r❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❝♦♠♣✉t❡s t❤❡ ncr(B) for any matrix space B ≤ Fn×n.

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆❧❣♦r✐t❤♠s ❢♦r ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦

• ✉r✈✐ts✱ ✷✵✵✹ ✿ ❉❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠s ❢♦r

✏❝♦♠♣r❡ss✐♦♥ s♣❛❝❡s✑

▼❛tr✐① s♣❛❝❡ B ✐s ❛ ❝♦♠♣r❡ss✐♦♥ s♣❛❝❡ ✐❢ rank(B) = ncr(B)✳

❚❤❡♦r❡♠ ✭●●❖❲ ✷✵✶✺✱ ■✈❛♥②♦s ❡t ❛❧✳✱✷✵✶✺ ✮ ❚❤❡r❡ ✐s ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ❝♦♠♣✉t❡s t❤❡ ncr(B) for any matrix space B ≤ Fn×n.

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ❈♦♠♠✉t❛t✐✈❡ r❛♥❦

❯s✐♥❣ rank(B) ≤ ncr(B) ≤ 2 · rank(B), ♦♥❡ ❣❡ts ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠s ❢♦r 1

2✲❛♣♣r♦①✐♠❛t✐♦♥ ♦❢

❈♦♠♠✉t❛t✐✈❡ r❛♥❦✳ ❚❤❡s❡ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ r❛♥❦ ❝♦♠♣✉t❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✇❡r❡ t❤❡ ♦♥❧② ❛❧❣♦r✐t❤♠s ✇❤✐❝❤ ❝♦♠♣✉t❡ ❛♥② ❝♦♥st❛♥t ❢❛❝t♦r ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❝♦♠♠✉t❛t✐✈❡ r❛♥❦✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❇❛s✐❝ Pr♦❜❧❡♠ ▼♦t✐✈❛t✐♦♥ Pr❡✈✐♦✉s ✇♦r❦

❆♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r ❈♦♠♠✉t❛t✐✈❡ r❛♥❦

▲❡❛❞s t♦ ❛ ♥❛t✉r❛❧ q✉❡st✐♦♥ ✇❤❡t❤❡r t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ ♦❢ 1

2 ❝❛♥ ❜❡ ✐♠♣r♦✈❡❞❄

❲❡ ❞❡✈✐s❡ ❛ ❞❡t❡r♠✐♥✐st✐❝ ♣♦❧②✲t✐♠❡ ❛❧❣♦r✐t❤♠ ✇❤✐❝❤ ✐♠♣r♦✈❡s t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦ t♦ 1 − ǫ ❢♦r ❛r❜✐tr❛r② ❝♦♥st❛♥t 0 < ǫ < 1✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ■❞❡❛

B = B1, B2, . . . , Bm ≤ Fn×n✳

B = x1B1 + x2B2 + . . . + xmBm ♦✈❡r t❤❡ ✜❡❧❞ F(x1, x2, . . . , xm)✳

❲❡ ❤❛✈❡ s♦♠❡ A ∈ B ✇✐t❤ s♦♠❡ r❛♥❦ r✳

❲❛♥t t♦ ✜♥❞ A′ ∈ B ✇✐t❤ rank(A′) > r✳

❲▲❖● ❛ss✉♠❡ A =      Ir . . . . . . ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ . . .     ✳ ❈♦♥s✐❞❡r t❤❡ ♠❛tr✐① A + B ∈ F(x1, x2, . . . , xm)n×n ✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ■❞❡❛

B = B1, B2, . . . , Bm ≤ Fn×n✳

B = x1B1 + x2B2 + . . . + xmBm ♦✈❡r t❤❡ ✜❡❧❞ F(x1, x2, . . . , xm)✳

❲❡ ❤❛✈❡ s♦♠❡ A ∈ B ✇✐t❤ s♦♠❡ r❛♥❦ r✳

❲❛♥t t♦ ✜♥❞ A′ ∈ B ✇✐t❤ rank(A′) > r✳

❲▲❖● ❛ss✉♠❡ A =      Ir . . . . . . ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ . . .     ✳ ❈♦♥s✐❞❡r t❤❡ ♠❛tr✐① A + B ∈ F(x1, x2, . . . , xm)n×n ✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ✐❞❡❛✭❈♦♥t✳✮

A + B = Ir + B11 B12 B21 B22

• ✳

❙✉♣♣♦s❡ B22 = 0 t❤❡♥ rank(A + B) = rank(B) ≤ 2r✳

rank(A) ✐s ❛❧r❡❛❞② 1

2✲❛♣♣r♦①✐♠❛t✐♦♥ of rank(B).

❖t❤❡r✇✐s❡ B22 = 0✱ c(x1, x2, . . . , xm) ❜❡ ❛ ♥♦♥✲③❡r♦ ❡♥tr② ♦❢ B22✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ✐❞❡❛✭❈♦♥t✳✮

A + B = Ir + B11 B12 B21 B22

• ✳

❙✉♣♣♦s❡ B22 = 0 t❤❡♥ rank(A + B) = rank(B) ≤ 2r✳

rank(A) ✐s ❛❧r❡❛❞② 1

2✲❛♣♣r♦①✐♠❛t✐♦♥ of rank(B).

❖t❤❡r✇✐s❡ B22 = 0✱ c(x1, x2, . . . , xm) ❜❡ ❛ ♥♦♥✲③❡r♦ ❡♥tr② ♦❢ B22✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ✐❞❡❛✭❈♦♥t✳✮

A + B = Ir + B11 B12 B21 B22

• ✳

❙✉♣♣♦s❡ B22 = 0 t❤❡♥ rank(A + B) = rank(B) ≤ 2r✳

rank(A) ✐s ❛❧r❡❛❞② 1

2✲❛♣♣r♦①✐♠❛t✐♦♥ of rank(B).

❖t❤❡r✇✐s❡ B22 = 0✱ c(x1, x2, . . . , xm) ❜❡ ❛ ♥♦♥✲③❡r♦ ❡♥tr② ♦❢ B22✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ✐❞❡❛✭❈♦♥t✳✮

❈♦♥s✐❞❡r t❤❡ ▼✐♥♦r M ♦❢ A + B ✇❤✐❝❤ ❤❛s c(x1, x2, . . . , xm) ❛s t❤❡ ❧❛st ❡♥tr②✳

M =      1 + ℓ11 ℓ12 . . . a1 ℓ21 1 + ℓ22 . . . a2 ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ b1 b2 . . . c(x1, x2, . . . , xm)     

(r+1)×(r+1)

det(M(x1, x2, . . . , xm)) = c(x1, x2, . . . , xm) + t❡r♠s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st ✷✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

▼❛✐♥ ✐❞❡❛✭❈♦♥t✳✮

❈♦♥s✐❞❡r t❤❡ ▼✐♥♦r M ♦❢ A + B ✇❤✐❝❤ ❤❛s c(x1, x2, . . . , xm) ❛s t❤❡ ❧❛st ❡♥tr②✳

M =      1 + ℓ11 ℓ12 . . . a1 ℓ21 1 + ℓ22 . . . a2 ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ b1 b2 . . . c(x1, x2, . . . , xm)     

(r+1)×(r+1)

det(M(x1, x2, . . . , xm)) = c(x1, x2, . . . , xm) + t❡r♠s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st ✷✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❋✐♥❛❧ ❙t❡♣

■❢ ✇❡ ❝❛♥ ✜♥❞ ❛ s❡tt✐♥❣ ♦❢ x = λ1✱x2 = λ2,. . .✱ xm = λm s✉❝❤ t❤❛t det(M(λ1, λ2, . . . , λm)) = 0✳

❚❤❡♥ ✇❡ ❣❡t ❛ r❛♥❦ r + 1 ♠❛tr✐① ✐♥ B✳ det(M(x1, x2, . . . , xm)) has degree 1 monomials.

❋❛❝t ■❢ ❛ ♥♦♥✲③❡r♦ ♣♦❧②♥♦♠✐❛❧ f(x1, x2, . . . , xm) ❤❛s ❛ ❞❡❣r❡❡ k ♠♦♥♦♠✐❛❧ ❛♥❞ deg(f) ≤ n✱ t❤❡♥ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ♥♦♥✲③❡r♦ ❛ss✐❣♥♠❡♥t x1 = λ1✱x2 = λ2,. . .✱ xm = λm ❢♦r f✱ ❜② tr②✐♥❣ O((mn)k) ❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ✏r❛♥❦ ✐♥❝r❡❛s✐♥❣ ❛ss✐❣♥♠❡♥t ♦❢ xi✬s✑ ❜② tr②✐♥❣ O(mn)

❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ♠❛tr✐① ♦❢ ❜✐❣❣❡r r❛♥❦ ✐♥ B✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❋✐♥❛❧ ❙t❡♣

■❢ ✇❡ ❝❛♥ ✜♥❞ ❛ s❡tt✐♥❣ ♦❢ x = λ1✱x2 = λ2,. . .✱ xm = λm s✉❝❤ t❤❛t det(M(λ1, λ2, . . . , λm)) = 0✳

❚❤❡♥ ✇❡ ❣❡t ❛ r❛♥❦ r + 1 ♠❛tr✐① ✐♥ B✳ det(M(x1, x2, . . . , xm)) has degree 1 monomials.

❋❛❝t ■❢ ❛ ♥♦♥✲③❡r♦ ♣♦❧②♥♦♠✐❛❧ f(x1, x2, . . . , xm) ❤❛s ❛ ❞❡❣r❡❡ k ♠♦♥♦♠✐❛❧ ❛♥❞ deg(f) ≤ n✱ t❤❡♥ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ♥♦♥✲③❡r♦ ❛ss✐❣♥♠❡♥t x1 = λ1✱x2 = λ2,. . .✱ xm = λm ❢♦r f✱ ❜② tr②✐♥❣ O((mn)k) ❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ✏r❛♥❦ ✐♥❝r❡❛s✐♥❣ ❛ss✐❣♥♠❡♥t ♦❢ xi✬s✑ ❜② tr②✐♥❣ O(mn)

❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ♠❛tr✐① ♦❢ ❜✐❣❣❡r r❛♥❦ ✐♥ B✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❋✐♥❛❧ ❙t❡♣

■❢ ✇❡ ❝❛♥ ✜♥❞ ❛ s❡tt✐♥❣ ♦❢ x = λ1✱x2 = λ2,. . .✱ xm = λm s✉❝❤ t❤❛t det(M(λ1, λ2, . . . , λm)) = 0✳

❚❤❡♥ ✇❡ ❣❡t ❛ r❛♥❦ r + 1 ♠❛tr✐① ✐♥ B✳ det(M(x1, x2, . . . , xm)) has degree 1 monomials.

❋❛❝t ■❢ ❛ ♥♦♥✲③❡r♦ ♣♦❧②♥♦♠✐❛❧ f(x1, x2, . . . , xm) ❤❛s ❛ ❞❡❣r❡❡ k ♠♦♥♦♠✐❛❧ ❛♥❞ deg(f) ≤ n✱ t❤❡♥ ♦♥❡ ❝❛♥ ✜♥❞ ❛ ♥♦♥✲③❡r♦ ❛ss✐❣♥♠❡♥t x1 = λ1✱x2 = λ2,. . .✱ xm = λm ❢♦r f✱ ❜② tr②✐♥❣ O((mn)k) ❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ✏r❛♥❦ ✐♥❝r❡❛s✐♥❣ ❛ss✐❣♥♠❡♥t ♦❢ xi✬s✑ ❜② tr②✐♥❣ O(mn)

❝❤♦✐❝❡s✳

• ✐✈❡s ❛ ♠❛tr✐① ♦❢ ❜✐❣❣❡r r❛♥❦ ✐♥ B✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❲❤❛t ✐❢ B22 = 0

B22 = 0 ✇❛s ♥❡❡❞❡❞ ❢♦r r❛♥❦ ✐♥❝r❡❛s❡✳ ❲❤❛t ✐❢ B22 = 0 ❄= ⇒ ❖♥❧② 1

2✲❛♣♣r♦①✐♠❛t✐♦♥✳

B22 = 0 made sure that det(M) has degree 1 monomials. ❲❤❛t ✐❢ ✇❡ ❧♦♦❦ ❢♦r ❞❡❣r❡❡ ✷ ♠♦♥♦♠✐❛❧s❄

❲❤❡♥ ❞♦❡s det(M) has degree two monomials?

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❲❤❛t ✐❢ B22 = 0

B22 = 0 ✇❛s ♥❡❡❞❡❞ ❢♦r r❛♥❦ ✐♥❝r❡❛s❡✳ ❲❤❛t ✐❢ B22 = 0 ❄= ⇒ ❖♥❧② 1

2✲❛♣♣r♦①✐♠❛t✐♦♥✳

B22 = 0 made sure that det(M) has degree 1 monomials. ❲❤❛t ✐❢ ✇❡ ❧♦♦❦ ❢♦r ❞❡❣r❡❡ ✷ ♠♦♥♦♠✐❛❧s❄

❲❤❡♥ ❞♦❡s det(M) has degree two monomials?

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❲❤❛t ✐❢ B22 = 0

B22 = 0 ✱ ❝♦♥s✐❞❡r ❛♥② (r + 1) × (r + 1) ♠✐♥♦r M ♦❢ A + B ✇✐t❤ Ir + B11 ❛❧✇❛②s ❜❡✐♥❣ t❤❡r❡✳ M =      1 + ℓ11 ℓ12 . . . a1 ℓ21 1 + ℓ22 . . . a2 ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ b1 b2 . . .     

(r+1)×(r+1)

▲❡♠♠❛ ■❢ B22 = 0 t❤❡♥ det(M(x1, x2, . . . , xm)) = − ∑r

i=1 aibi + t❡r♠s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st ✸✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❲❤❛t ✐❢ B22 = 0

B22 = 0 ✱ ❝♦♥s✐❞❡r ❛♥② (r + 1) × (r + 1) ♠✐♥♦r M ♦❢ A + B ✇✐t❤ Ir + B11 ❛❧✇❛②s ❜❡✐♥❣ t❤❡r❡✳ M =      1 + ℓ11 ℓ12 . . . a1 ℓ21 1 + ℓ22 . . . a2 ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ b1 b2 . . .     

(r+1)×(r+1)

▲❡♠♠❛ ■❢ B22 = 0 t❤❡♥ det(M(x1, x2, . . . , xm)) = − ∑r

i=1 aibi + t❡r♠s ♦❢ ❞❡❣r❡❡ ❛t ❧❡❛st ✸✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

2 3✲❛♣♣r♦①✐♠❛t✐♦♥

■❢ ❞❡❣r❡❡ t✇♦ t❡r♠s ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ M ❛r❡ ③❡r♦ t❤❡♥

B21B12 = 0 B22 = 0

▲❡♠♠❛ ❆❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t rank(B) ≤ 3

2r✳

Pr♦♦❢✳ ■❢ rank(B12) ≤ r

2 t❤❡♥ tr✐✈✐❛❧✳ ❖t❤❡r✇✐s❡ r❛♥❦ rank(B21) ≤ r 2 ❜②

r❛♥❦✲♥✉❧❧✐t② t❤❡♦r❡♠✳ ❊✐t❤❡r ✇❛②✱ rank(B) ≤ 3

2r✳

❚❤✉s ✐❢ ♥♦ ❞❡❣r❡❡ ✷ t❡r♠s t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❛❧r❡❛❞②

❖t❤❡r✇✐s❡ ✐♥❝r❡❛s❡ t❤❡ r❛♥❦ ❜② tr②✐♥❣ O((mn)2) ❝❤♦✐❝❡s✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

2 3✲❛♣♣r♦①✐♠❛t✐♦♥

■❢ ❞❡❣r❡❡ t✇♦ t❡r♠s ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ M ❛r❡ ③❡r♦ t❤❡♥

B21B12 = 0 B22 = 0

▲❡♠♠❛ ❆❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t rank(B) ≤ 3

2r✳

Pr♦♦❢✳ ■❢ rank(B12) ≤ r

2 t❤❡♥ tr✐✈✐❛❧✳ ❖t❤❡r✇✐s❡ r❛♥❦ rank(B21) ≤ r 2 ❜②

r❛♥❦✲♥✉❧❧✐t② t❤❡♦r❡♠✳ ❊✐t❤❡r ✇❛②✱ rank(B) ≤ 3

2r✳

❚❤✉s ✐❢ ♥♦ ❞❡❣r❡❡ ✷ t❡r♠s t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❛❧r❡❛❞②

❖t❤❡r✇✐s❡ ✐♥❝r❡❛s❡ t❤❡ r❛♥❦ ❜② tr②✐♥❣ O((mn)2) ❝❤♦✐❝❡s✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

2 3✲❛♣♣r♦①✐♠❛t✐♦♥

■❢ ❞❡❣r❡❡ t✇♦ t❡r♠s ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ M ❛r❡ ③❡r♦ t❤❡♥

B21B12 = 0 B22 = 0

▲❡♠♠❛ ❆❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t rank(B) ≤ 3

2r✳

Pr♦♦❢✳ ■❢ rank(B12) ≤ r

2 t❤❡♥ tr✐✈✐❛❧✳ ❖t❤❡r✇✐s❡ r❛♥❦ rank(B21) ≤ r 2 ❜②

r❛♥❦✲♥✉❧❧✐t② t❤❡♦r❡♠✳ ❊✐t❤❡r ✇❛②✱ rank(B) ≤ 3

2r✳

❚❤✉s ✐❢ ♥♦ ❞❡❣r❡❡ ✷ t❡r♠s t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❛❧r❡❛❞②

❖t❤❡r✇✐s❡ ✐♥❝r❡❛s❡ t❤❡ r❛♥❦ ❜② tr②✐♥❣ O((mn)2) ❝❤♦✐❝❡s✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❉❡❣r❡❡ ✸ t❡r♠s

❲❡ s❛✇ t❤❛t ✐❢ ❞❡❣r❡❡ ♦♥❡ ❛♥❞ ❞❡❣r❡❡ t✇♦ t❡r♠s ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ M ❛r❡ ③❡r♦ t❤❡♥

B21B12 = 0 B22 = 0

❲❤❛t ✐❢ ❞❡❣r❡❡ t❤r❡❡ t❡r♠s ❛r❡ ❛❧s♦ ③❡r♦❄ ▲❡♠♠❛ ■❢ ❞❡❣r❡❡ ✶✱✷ ❛♥❞ ✸ t❡r♠s ❛r❡ ❛❧❧ ③❡r♦ ✐♥ det(M) ❢♦r ❛❧❧ ▼ t❤❡♥ B22 = 0✱ B21B12 = 0 ❛♥❞ B21B11B12 = 0✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

3 4✲❛♣♣r♦①✐♠❛t✐♦♥

▲❡♠♠❛ ❆❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t rank(B) ≤ 4

3r✳

❚❤✉s ✐❢ ♥♦ ❞❡❣r❡❡ ✶✱✷✱✸ t❡r♠s t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❛❧r❡❛❞②✳

❖t❤❡r✇✐s❡ ✐♥❝r❡❛s❡ t❤❡ r❛♥❦ ❜② tr②✐♥❣ O((mn)3) ❝❤♦✐❝❡s✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

3 4✲❛♣♣r♦①✐♠❛t✐♦♥

▲❡♠♠❛ ❆❜♦✈❡ ❝♦♥❞✐t✐♦♥s ✐♠♣❧② t❤❛t rank(B) ≤ 4

3r✳

❚❤✉s ✐❢ ♥♦ ❞❡❣r❡❡ ✶✱✷✱✸ t❡r♠s t❤❡♥ ✇❡ ❛r❡ ❞♦♥❡ ❛❧r❡❛❞②✳

❖t❤❡r✇✐s❡ ✐♥❝r❡❛s❡ t❤❡ r❛♥❦ ❜② tr②✐♥❣ O((mn)3) ❝❤♦✐❝❡s✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

• ❡♥❡r❛❧✐③✐♥❣ ❛❜♦✈❡ ✐❞❡❛s

❲❡ ❤❛✈❡ s♦♠❡ A ∈ B✱ ✇✐t❤ rank(A) = r✳ ❆❜♦✈❡ ❞✐s❝✉ss✐♦♥ ❤✐♥ts t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡✳ ❈♦♥❥❡❝t✉r❡ ❋♦r ❛♥② k ≤ n✱ ❡✐t❤❡r rank(B) ≤ r

• 1 + 1

k

• ♦r ✇❡ ❝❛♥ ✐♥❝r❡❛s❡ t❤❡

r❛♥❦ ❜② tr②✐♥❣ O((mn)k) ❝❤♦✐❝❡s✳ ❲❡ ♣r♦✈❡ t❤✐s ❝♦♥❥❡❝t✉r❡ ❜② s♦ ❝❛❧❧❡❞ ✏❲♦♥❣ ❙❡q✉❡♥❝❡s✑✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

• ❡♥❡r❛❧✐③✐♥❣ ❛❜♦✈❡ ✐❞❡❛s

❲❡ ❤❛✈❡ s♦♠❡ A ∈ B✱ ✇✐t❤ rank(A) = r✳ ❆❜♦✈❡ ❞✐s❝✉ss✐♦♥ ❤✐♥ts t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❥❡❝t✉r❡✳ ❈♦♥❥❡❝t✉r❡ ❋♦r ❛♥② k ≤ n✱ ❡✐t❤❡r rank(B) ≤ r

• 1 + 1

k

• ♦r ✇❡ ❝❛♥ ✐♥❝r❡❛s❡ t❤❡

r❛♥❦ ❜② tr②✐♥❣ O((mn)k) ❝❤♦✐❝❡s✳ ❲❡ ♣r♦✈❡ t❤✐s ❝♦♥❥❡❝t✉r❡ ❜② s♦ ❝❛❧❧❡❞ ✏❲♦♥❣ ❙❡q✉❡♥❝❡s✑✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❋✐♥❛❧ ❛❧❣♦r✐t❤♠

❙❡t k = O 1

ǫ

• ❛♥❞ ✇❡ ❣❡t t❤❡ ❞❡s✐r❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦✳

❘✉♥♥✐♥❣ t✐♠❡ ✐s nO( 1

ǫ)✳

❲❡ ❛❧s♦ s❤♦✇ t✐❣❤t ❡①❛♠♣❧❡s ✇❤❡r❡ t❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ❣✐✈❡ ❜❡tt❡r t❤❛♥ (1 − ǫ) ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦✳

❙♦ ❛♥❛❧②s✐s ❛❜♦✈❡ ✐s t✐❣❤t✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❋✐♥❛❧ ❛❧❣♦r✐t❤♠

❙❡t k = O 1

ǫ

• ❛♥❞ ✇❡ ❣❡t t❤❡ ❞❡s✐r❡❞ ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦✳

❘✉♥♥✐♥❣ t✐♠❡ ✐s nO( 1

ǫ)✳

❲❡ ❛❧s♦ s❤♦✇ t✐❣❤t ❡①❛♠♣❧❡s ✇❤❡r❡ t❤✐s ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ❣✐✈❡ ❜❡tt❡r t❤❛♥ (1 − ǫ) ❛♣♣r♦①✐♠❛t✐♦♥ r❛t✐♦✳

❙♦ ❛♥❛❧②s✐s ❛❜♦✈❡ ✐s t✐❣❤t✳

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦

■♥tr♦❞✉❝t✐♦♥ ▼❛✐♥ ❛❧❣♦r✐t❤♠ ❆ s✐♠♣❧❡ 1

2 ✲❛♣♣r♦①✐♠❛t✐♦♥ ❛❧❣♦r✐t❤♠

■❞❡❛s ❢♦r ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥

❚❤❛♥❦s

❚❤❛♥❦s ❢♦r ❧✐st❡♥✐♥❣

▼❛r❦✉s ❇❧äs❡r✱ ●♦r❛✈ ❏✐♥❞❛❧ ❛♥❞ ❆♥✉r❛❣ P❛♥❞❡② ❉❡t❡r♠✐♥✐st✐❝ P❚❆❙ ❢♦r ❈♦♠♠✉t❛t✐✈❡ ❘❛♥❦