Examples Key observation: There are just ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sums of terms of the form ℓ 1 ··· ℓ d . � d � linearly independent k -th order k partial derivatives of ℓ 1 ··· ℓ d .
Examples Key observation: There are just For a generic polynomial, you would all partial derivatives to be linearly independent. ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sums of terms of the form ℓ 1 ··· ℓ d . � d � linearly independent k -th order k partial derivatives of ℓ 1 ··· ℓ d .
Examples Key observation: There are just For a generic polynomial, you would all partial derivatives to be linearly independent. ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sums of terms of the form ℓ 1 ··· ℓ d . � d � linearly independent k -th order k partial derivatives of ℓ 1 ··· ℓ d . �∏ � ∂ = k ( ℓ 1 ··· ℓ d ) ⊆ span ℓ i : S ⊆ [ d ] , | S | = d − k i ∈ S
Examples For a generic polynomial, you would all partial derivatives to be linearly independent. Key observation: There are just ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sums of terms of the form ℓ 1 ··· ℓ d . � d � linearly independent k -th order k partial derivatives of ℓ 1 ··· ℓ d . �∏ � ∂ = k ( ℓ 1 ··· ℓ d ) ⊆ span ℓ i : S ⊆ [ d ] , | S | = d − k i ∈ S For f = Det d , the symbolic determinant of a d × d matrix, we have � 2 linearly independent ( d − k ) × ( d − k ) minors. � d k
Examples For a generic polynomial, you would all partial derivatives to be linearly independent. . Key observation: There are just ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sums of terms of the form ℓ 1 ··· ℓ d . � d � linearly independent k -th order k partial derivatives of ℓ 1 ··· ℓ d . �∏ � ∂ = k ( ℓ 1 ··· ℓ d ) ⊆ span ℓ i : S ⊆ [ d ] , | S | = d − k i ∈ S For f = Det d , the symbolic determinant of a d × d matrix, we have � 2 linearly independent ( d − k ) × ( d − k ) minors. � d k � d Therefore, if Det d = ∑ s i = 1 ℓ i 1 ··· ℓ id , then s ≥ � d / 2
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d .
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . m ∂ x α ∂ = k coefg. of m in ∂ x α ( f ) Mons of degree d − k
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d .
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d .
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . ∂ x ( Q 1 ··· Q r ) = ∂ x ( Q 1 ) · Q 2 ··· Q r + ··· + Q 1 ··· Q r − 1 · ∂ x ( Q r )
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . ∂ x ( Q 1 ··· Q r ) = ∂ x ( Q 1 ) · Q 2 ··· Q r + ··· + Q 1 ··· Q r − 1 · ∂ x ( Q r )
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . � � � d · x = ∏ ∂ x ( Q 1 ··· Q r ) Q i : S ⊂ [ r ] , | S | = r − 1 = span i ∈ S
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . � � � d · ∂ = k ( Q 1 ··· Q r ) x = k ∏ Q i : S ⊂ [ r ] , | S | = r − k = span i ∈ S
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . � � � d · ∂ = k ( Q 1 ··· Q r ) x = k ∏ Q i : S ⊂ [ r ] , | S | = r − k = span i ∈ S Key observation: Many low-degree combinations of partial derivatives are zero if all Q i s have low degree.
Examples Key observation: There are “few” linearly independent partial Key observation: Many low-degree combinations of partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree.
Examples Key observation: There are “few” linearly independent partial Key observation: Many low-degree combinations of partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. � � x = ℓ ∂ = k ( f ) Γ ( f ) = dim Dimension of shifted partial derivatives
Examples Key observation: There are “few” linearly independent partial ▶ [Nisan-Wigderson-95]: ΣΠ d Σ circuits, sum of terms of the form ℓ 1 ··· ℓ d . derivatives of ℓ 1 ··· ℓ d . � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . m x β ∂ x α x = ℓ ∂ = k coefg. of m in x β ∂ x α ( f ) Mons of degree ℓ + d − k
Key observation: Many low-degree combinations of partial Examples... � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree.
Key observation: Many low-degree combinations of partial Examples... � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d .
Key observation: Many low-degree combinations of partial Examples... � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d .
Key observation: Many low-degree combinations of partial Examples... Low degree mons. High degree mons. � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d .
Examples... Low degree [GKKS-12] mons. High degree mons. Key observation: Many low-degree combinations of partial � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓
Examples... Low degree [GKKS-12] small support mons. High degree large support mons. High degree mons. Key observation: Many low-degree combinations of partial � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2
Examples... Low degree [GKKS-12] small support mons. High degree large support mons. mons. High degree Key observation: Many low-degree combinations of partial variables to zero � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2 ▶ Idea 1 - Random restrictions: Randomly set a small number of
Examples... Low degree [GKKS-12] small support mons. High degree large support mons. mons. High degree Key observation: Many low-degree combinations of partial variables to zero � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2 ▶ Idea 1 - Random restrictions: Randomly set a small number of
Examples... Low degree variables to zero [GKKS-12] small support mons. High degree High degree mons. large support mons. monomials Key observation: Many low-degree combinations of partial � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2 ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear
Examples... Low degree variables to zero [GKKS-12] small support mons. High degree High degree mons. large support mons. monomials Key observation: Many low-degree combinations of partial � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2 ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear
Examples... Low degree variables to zero [GKKS-12] small support mons. High degree large support mons. mons. High degree monomials Key observation: Many low-degree combinations of partial � � d ΣΠ d circuits, terms of the ▶ [Gupta-Kamath-Kayal-S-13]: ΣΠ form Q 1 ··· Q � d . derivatives are zero if all Q i s have low degree. ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ✓ Eg. x d / 2 x d / 2 Eg. x 1 ··· x d 1 2 ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear
Examples... variables to zero monomials ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear
Examples... variables to zero monomials ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear dim ( x = ℓ ∂ = k ( f )) Γ ( f ) = Dimension of shifted partials of f .
Examples... variables to zero monomials ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear dim ( x = ℓ ∂ = k ( ρ ( f ))) Γ ( f ) = Dimension of shifted partials of a random restriction of f .
Examples... variables to zero monomials ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . ▶ Idea 1 - Random restrictions: Randomly set a small number of ▶ Idea 2 - Multilinear projection: Discard all non-multilinear dim ( mult ◦ x = ℓ ∂ = k ( ρ ( f ))) Γ ( f ) = Dimension of projected shifted partials of a random restriction of f .
Examples... ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . dim ( mult ◦ x = ℓ ∂ = k ( ρ ( f ))) Γ ( f ) = Dimension of projected shifted partials of a random restriction of f .
Examples... ▶ [Kayal-Limaye-Saha-Srinivasan-13], [Kumar-Saraf-13]: hom. ΣΠΣΠ circuits: terms like Q 1 ··· Q b with total degree d . dim ( mult ◦ x = ℓ ∂ = k ( ρ ( f ))) Γ ( f ) = Dimension of projected shifted partials of a random restriction of f . m x β ∂ x α x = ℓ ∂ = k coefg. of m in x β ∂ x α ( ρ ( f )) Multilinear mons of degree ℓ + d − k
PSPD for hom. depth- circuits. PSPD for a generic depth- circuit ? We already have a complexity measure How large is Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know Handling depth- 5 circuits
PSPD for a generic depth- circuit ? How large is Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know Handling depth- 5 circuits We already have a complexity measure Γ PSPD for hom. depth- 4 circuits.
Small a lower bound against depth- circuits. Large separation between depth- and depth- circuits. ... still don’t know Handling depth- 5 circuits We already have a complexity measure Γ PSPD for hom. depth- 4 circuits. How large is Γ PSPD for a generic depth- 5 circuit ?
Large separation between depth- and depth- circuits. ... still don’t know Handling depth- 5 circuits We already have a complexity measure Γ PSPD for hom. depth- 4 circuits. How large is Γ PSPD for a generic depth- 5 circuit ? Small ⇒ a lower bound against depth- 5 circuits.
... still don’t know Handling depth- 5 circuits We already have a complexity measure Γ PSPD for hom. depth- 4 circuits. How large is Γ PSPD for a generic depth- 5 circuit ? Small ⇒ a lower bound against depth- 5 circuits. Large ⇒ separation between depth- 5 and depth- 4 circuits.
... still don’t know Handling depth- 5 circuits We already have a complexity measure Γ PSPD for hom. depth- 4 circuits. How large is Γ PSPD for a generic depth- 5 circuit ? Small ⇒ a lower bound against depth- 5 circuits. Large ⇒ separation between depth- 5 and depth- 4 circuits.
Evaluating the complexity measure � ∂ = k ( f ) � Γ k ( f ) = dim
Evaluating the complexity measure m ∂ x α ∂ = k coefg. of m in ∂ x α ( f ) Monomials of degree d − k
Evaluating the complexity measure eval. of m ∂ x α ∂ = k coefg. of m in ∂ x α ( f ) Monomials of degree d − k a ¯ ∂ x α ∂ = k ∂ x α ( ρ ( f )) at ¯ a Points in � n q
Evaluating the complexity measure eval. of Small rank Small rank m ∂ x α ∂ = k coefg. of m in ∂ x α ( f ) Monomials of degree d − k ⇓ a ¯ ∂ x α ∂ = k ∂ x α ( ρ ( f )) at ¯ a Points in � n q
[Grigoriev-Karpinski] f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s
[Grigoriev-Karpinski] Low degree terms. High degree terms. f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s
[Grigoriev-Karpinski] Low degree terms. High degree High degree f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s high rank terms low rank terms Eg. ℓ d / 3 ℓ d / 3 ( ℓ 1 + 3 ℓ 2 ) d / 3 1 2
[Grigoriev-Karpinski] Low degree terms. High degree High degree [NW-95] f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s high rank terms low rank terms ✓ Eg. ℓ d / 3 ℓ d / 3 ( ℓ 1 + 3 ℓ 2 ) d / 3 1 2
[Grigoriev-Karpinski] [NW-95] Low degree terms. High degree High degree [NW-95] f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s high rank terms low rank terms ✓ Eg. ℓ d / 3 ℓ d / 3 ( ℓ 1 + 3 ℓ 2 ) d / 3 1 2 ✓
[Grigoriev-Karpinski] High degree Observation [NW-95] [NW-95] zero. High degree terms. Low degree f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s high rank terms low rank terms ✓ Eg. ℓ d / 3 ℓ d / 3 ( ℓ 1 + 3 ℓ 2 ) d / 3 1 2 ✓ If dim { ℓ 1 , ··· , ℓ r } is large, then almost all evaluations of it on � n q are
[Grigoriev-Karpinski] High degree Observation [NW-95] [NW-95] zero. High degree terms. Low degree f = ℓ 11 ··· ℓ 1 d 1 + ··· + ℓ s 1 ··· ℓ sd s high rank terms low rank terms ✓ Eg. ℓ d / 3 ℓ d / 3 ( ℓ 1 + 3 ℓ 2 ) d / 3 1 2 ✓ If dim { ℓ 1 , ··· , ℓ r } is large, then almost all evaluations of it on � n q are
[Grigoriev-Karpinski] m ∂ x α ∂ = k coefg. of m in ∂ x α ( f ) Mons. of degree d − k
[Grigoriev-Karpinski] a ¯ ∂ x α ∂ = k eval. of ∂ x α ( f ) at ¯ a � n q
[Grigoriev-Karpinski] a ¯ ∂ x α ∂ = k eval. of ∂ x α ( f ) at ¯ a � n q
[Grigoriev-Karpinski] Lemma matrix has small rank when a certain small set of columns are removed. a ¯ ∂ x α ∂ = k eval. of ∂ x α ( f ) at ¯ a � n q If f is computable by a small ΣΠΣ circuit over � q , then there the above
[Grigoriev-Karpinski] Lemma matrix has small rank when a certain small set of columns are removed. Lemma removed only few columns. a ¯ ∂ x α ∂ = k eval. of ∂ x α ( f ) at ¯ a � n q If f is computable by a small ΣΠΣ circuit over � q , then there the above For Det n or Perm n the above matrix remains full rank, as long as we
Lifting to depth five Types of products of linear polynomials: Low degree products. High degree products. [GKKS] ΣΠΣΠΣ
Lifting to depth five Types of products of linear polynomials: Low degree products. High degree products. [GKKS] ΣΠΣΠΣ ✓
Lifting to depth five Types of products of linear polynomials: Low degree products. High degree, large rank products. High degree, small rank products. [GKKS] ΣΠΣΠΣ ✓ Eg. ℓ d / 2 ℓ d / 2 Eg. ℓ 1 ··· ℓ d 1 2
Lifting to depth five large rank products. [GKKS] small rank products. High degree, High degree, products. Low degree Types of products of linear polynomials: ΣΠΣΠΣ ✓ Eg. ℓ d / 2 ℓ d / 2 Eg. ℓ 1 ··· ℓ d 1 2 ✓
Lifting to depth five High degree, Observation [GKKS] small rank products. zero. large rank products. High degree, products. Low degree Types of products of linear polynomials: ΣΠΣΠΣ ✓ Eg. ℓ d / 2 ℓ d / 2 Eg. ℓ 1 ··· ℓ d 1 2 ✓ If dim { ℓ 1 , ··· , ℓ r } is large, then almost all evaluations of it on � n q are
Lifting to depth five High degree, Observation [GKKS] small rank products. zero. large rank products. High degree, products. Low degree Types of products of linear polynomials: ΣΠΣΠΣ ✓ Eg. ℓ d / 2 ℓ d / 2 Eg. ℓ 1 ··· ℓ d 1 2 ✓ If dim { ℓ 1 , ··· , ℓ r } is large, then almost all evaluations of it on � n q are
Rank of the eval. version of PSPD We know this rank is large: m x β ∂ x α x = ℓ ∂ = k coefg. of m in x β ∂ x α ( f ) Mons of degree ℓ + d − k
Rank of the eval. version of PSPD We know this rank is large: Need to show this rank is large: m x β ∂ x α x = ℓ ∂ = k coefg. of m in x β ∂ x α ( f ) Mons of degree ℓ + d − k a x β ∂ x α x = ℓ ∂ = k eval. of x β ∂ x α ( f ) at a { 0,1 } n
Switching to the evaluation perspective Large rank Vandermonde [KLSS,KS] Large rank a x β ∂ x α Mons of degree ℓ + d − k � n q a x β ∂ x α = x = ℓ ∂ = k eval. of x β ∂ x α ( f ) at a � n q
Switching to the evaluation perspective Large rank Vandermonde Large rank a x β ∂ x α Mons of degree ℓ + d − k ∵ [KLSS,KS] � n q a x β ∂ x α = x = ℓ ∂ = k eval. of x β ∂ x α ( f ) at a � n q
Switching to the evaluation perspective Large rank Large rank a x β ∂ x α ∵ Vandermonde Mons of degree ℓ + d − k ∵ [KLSS,KS] � n q a x β ∂ x α = x = ℓ ∂ = k eval. of x β ∂ x α ( f ) at a � n q
Switching to the evaluation perspective Large rank Large rank a x β ∂ x α ∵ Vandermonde Mons of degree ℓ + d − k ∵ [KLSS,KS] � n q a x β ∂ x α = x = ℓ ∂ = k eval. of x β ∂ x α ( f ) at a � n q
Issues to be resolved . Fix: Prove a really good rank lower bound on the left matrix. Vandermonde is slightly tall. the matrix is still slightly fat and the Issue 3: Even with . for some random Fix: Ok fine. Work with , is never zero over Issue 1: [Fat matrix] , over Issue 2: But then . Fix: Make the matrix slimmer by only considering evaluations on rank. all matrix] could be zero, even if both are full [T (Barely manages to work for a specific explicit polynomial. Phew!)
Issues to be resolved Fix: Ok fine. Work with Fix: Prove a really good rank lower bound on the left matrix. Vandermonde is slightly tall. the matrix is still slightly fat and the Issue 3: Even with . for some random . , is never zero over , over Issue 2: But then . Fix: Make the matrix slimmer by only considering evaluations on rank. all matrix] could be zero, even if both are full (Barely manages to work for a specific explicit polynomial. Phew!) Issue 1: [Fat matrix] × [T
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