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On the Uniqueness of Simultaneous Rational Function Reconstruction

Ilaria Zappatore a joint work with E. Guerrini, R. Lebreton LIRMM, Université de Montpellier, CNRS

Journées Nationales de Calcul Formel 2020 CIRM, Luminy.

The Simultaneous Rational Function Reconstruction (SRFR)

Rational Function Reconstruction

Rational Function Reconstruction Instance: a, u ∈ K[x] with deg(u) < deg(a), and N, D ∈ N Solution: (v, d) ∈ K[x]2 such that

• v

d ≡ u mod a

v du a

• deg(v) < N,
• deg(d) < D.

Consider the homogeneous linear system equations unknowns 1 a non trivial solution a N D 1 unique solution: v d pvmin pdmin for p x .

1

Rational Function Reconstruction

Rational Function Reconstruction Instance: a, u ∈ K[x] with deg(u) < deg(a), and N, D ∈ N Solution: (v, d) ∈ K[x]2 such that

• v

d ≡ u mod a ⇒ v ≡ du mod a

v du a

• deg(v) < N,
• deg(d) < D.

Consider the homogeneous linear system equations unknowns 1 a non trivial solution a N D 1 unique solution: v d pvmin pdmin for p x .

1

Rational Function Reconstruction

Rational Function Reconstruction Instance: a, u ∈ K[x] with deg(u) < deg(a), and N, D ∈ N Solution: (v, d) ∈ K[x]2 such that

• v ≡ du mod a
• deg(v) < N,
• deg(d) < D.

RFR is

• Padé Approximation: a = xf,
• Cauchy interpolation: a = f

i=1(x − αi)

Consider the homogeneous linear system equations unknowns 1 a non trivial solution a N D 1 unique solution: v d pvmin pdmin for p x .

1

Rational Function Reconstruction

Rational Function Reconstruction Instance: a, u ∈ K[x] with deg(u) < deg(a), and N, D ∈ N Solution: (v, d) ∈ K[x]2 such that

• v ≡ du mod a
• deg(v) < N,
• deg(d) < D.

Consider the homogeneous linear system #equations = #unknowns − 1

• =

⇒ ∃ a non trivial solution ↓ ↓ deg(a) N + D − 1 unique solution: v d pvmin pdmin for p x .

1

Rational Function Reconstruction

Rational Function Reconstruction Instance: a, u ∈ K[x] with deg(u) < deg(a), and N, D ∈ N Solution: (v, d) ∈ K[x]2 such that

• v ≡ du mod a
• deg(v) < N,
• deg(d) < D.

Consider the homogeneous linear system #equations = #unknowns − 1

• =

⇒ ∃ a non trivial solution ↓ ↓ deg(a) N + D − 1 unique solution: (v, d) = (pvmin, pdmin) for p ∈ K[x].

1

Vector Rational function Reconstruction

Vector Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni, Di∈ Nn Solution: (vi, di) such that

• vi≡ diuimodai,
• deg(vi) < Ni,
• deg(di) < Di.

Apply RFR component-wise #equations = #unknowns − 1

• =

⇒ ∃ a non trivial solution ↓ ↓ ai Ni + Di − 1 unique solution: (vi, di) = (pvmin,i, pdmin,i) for p ∈ K[x] .

2

Simultaneous Rational function Reconstruction

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D∈ N Solution: (vi,d) such that

• vi ≡ d ui mod ai,
• deg(vi) < Ni,
• deg(d) < D.

RFR ai Ni n D 1 ai Ni D 1 SRFR existence, uniqueness existence, uniqueness?

3

Simultaneous Rational function Reconstruction

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D∈ N Solution: (vi,d) such that

• vi ≡ d ui mod ai,
• deg(vi) < Ni,
• deg(d) < D.

Use the common denominator property, #equations = #unknowns − 1 → existence, not uniqueness ↓ ↓ deg(ai) (Ni) + D − 1

RFR ai Ni n D 1 ai Ni D 1 SRFR existence, uniqueness existence, uniqueness?

3

Simultaneous Rational function Reconstruction

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D∈ N Solution: (vi,d) such that

• vi ≡ d ui mod ai,
• deg(vi) < Ni,
• deg(d) < D.

RFR deg(ai) = (Ni) + n(D − 1) ≤ deg(ai) = (Ni) + D − 1 SRFR existence, uniqueness existence, uniqueness?

3

Our Result

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai,
• deg(vi) < Ni,
• deg(d) < D.

Theorem, [Guerrini, Lebreton, Z.] If n

i=1 deg(ai) = n i=1 Ni + D − 1,then

for almost all instances u ⇒ uniqueness.

4

SRFR Applications

SRFR Applications

SRFR APPLICATIONS Polynomial Linear System

reconstruct y = A−1b ∈ K(x)n×1 given evaluations Cramer’s rule → common denominator uniqueness→ unique reconstruction less points → lower complexity

Decoding Interleaved Reed-Solomon Codes

recovering f ∈ K[x]n×1 given evaluations, some erroneous error locator polynomial → common denominator uniqueness → unique decoding less points → more errors

5

Decoding IRS codes

f(𝝱1) … f(𝝱3) f(x) f(𝝱n) deg(f)<k received matrix

Decoding IRS ← → SRFR

Instance: the received matrix (ui,j) of an IRS(k, n) → interpolators Ui(x) Solution: (v, d) such that

• vi ≡ dUi mod n

j=1(x − αj) ⇐

⇒ vi(αj) = d(αj)ui,j,

• deg(d) < e + 1,
• deg(vi) < k + e + 1.

f is a solution,

j err x j error locator polynomial

6

Decoding IRS codes

f(𝝱1) … f(𝝱3) f(x) f(𝝱n) deg(f)<k received matrix

Decoding IRS ← → SRFR

Instance: the received matrix (ui,j) of an IRS(k, n) → interpolators Ui(x) Solution: (v, d) such that

• Λfi ≡ ΛUi mod n

j=1(x − αj) ⇐

⇒ Λ(αj)f(αj) = Λ(αj)ui,j,

• deg(d) < e + 1,
• deg(vi) < k + e + 1.

(Λf, Λ) is a solution, Λ =

j err (x − αj) error locator polynomial

6

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai

⇐ ⇒ (v, d)

• Id

−u

• R

≡ 0 mod (0, . . . , ai, . . . , 0)i

• M

⇐ ⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• vi

Ni

• d

D What about the degree constraints? row degrees

7

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai

⇐ ⇒ (v, d)

• Id

−u

• R

≡ 0 mod (0, . . . , ai, . . . , 0)i

• M

⇐ ⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• deg(vi) < Ni,

vi Ni

• deg(d) < D

d D What about the degree constraints? row degrees

7

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai

⇐ ⇒ (v, d)

• Id

−u

• R

≡ 0 mod (0, . . . , ai, . . . , 0)i

• M

⇐ ⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• deg(vi) < Ni,
• deg(d) < D

What about the degree constraints? row degrees

7

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai

⇐ ⇒ (v, d)

• Id

−u

• R

≡ 0 mod (0, . . . , ai, . . . , 0)i

• M

⇐ ⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• deg(vi) < Ni,
• deg(d) < D

What about the degree constraints? → row degrees

7

Row degrees and Ordered Weak Popov basis

p x3 + 2 2x2 + 5 3x + 2 deg 3 2 1 → rdeg(p) = max(deg(pi)) =3 s 1 2 1 degs 4 4 2 → rdegs(p) = max(degs(pi)) = 4 p x3 2 2x2 5 3x 2 deg 3 2 1 rdeg p pi 3 s 1 2 1 degs 4 4 2 rdegs p

s pi

4 pivot degree

s p2

2

Ordered Weak Popov form

Defjnition: The pivots are on the diagonal. Example: P x3 1 3x 1 x 2x 2x3 4 x3 2x x2 1 x2 2x4 x4 3x 5 x6 1 3 0 2 1 2 1 3 0 4 1 3 1 2 0 5 1 7 1 Ordered weak Popov Basis row degrees of the module uniquely defjned

8

Row degrees and Ordered Weak Popov basis

p x3 + 2 2x2 + 5 3x + 2 deg 3 2 1 → rdeg(p) = max(deg(pi)) = 3 s 1 2 1 degs 4 4 2 → rdegs(p) = max(degs(pi)) = 4 pivot degree = degs(p2) = 2

Ordered Weak Popov form

Defjnition: The pivots are on the diagonal. Example: P x3 1 3x 1 x 2x 2x3 4 x3 2x x2 1 x2 2x4 x4 3x 5 x6 1 3 0 2 1 2 1 3 0 4 1 3 1 2 0 5 1 7 1 Ordered weak Popov Basis row degrees of the module uniquely defjned

8

Row degrees and Ordered Weak Popov basis

p x3 + 2 2x2 + 5 3x + 2 deg 3 2 1 → rdeg(p) = max(deg(pi)) = 3 s 1 2 1 degs 4 4 2 → rdegs(p) = max(degs(pi)) = 4 pivot degree = degs(p2) = 2

Ordered Weak Popov form

Defjnition: The pivots are on the diagonal. Example: P =    x3 + 1 3x + 1 x + 2x 2x3 + 4 x3 + 2x x2 + 1 x2 + 2x4 x4 + 3x + 5 x6 + 1    =         [3]0 [2]1 [2]1 [3]0 [4]1 [3]1 [2]0 [5]1 [7]1         Ordered weak Popov Basis ↔ row degrees of the module uniquely defjned

8

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction

Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai ⇐

⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• 

  deg(vi) < Ni deg(d) < D ⇐ ⇒ rdegs(vi, d) < 0, where s = (−N1, . . . , −Nn, −D),

Theorem, [Guerrini, Lebreton, Z.]

If

n i 1

ai

n i 1 Ni

D 1, then the generic row degree of the relation module is 0 1 .

9

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction

Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai ⇐

⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• 

  deg(vi) < Ni deg(d) < D ⇐ ⇒ rdegs(vi, d) < 0, where s = (−N1, . . . , −Nn, −D),

• solution space SRFR ←

→

• rdered weak Popov basis of A,

negative row degree

• row degree of the form (0, . . . , 0, −1) ⇒ uniqueness

Theorem, [Guerrini, Lebreton, Z.]

If

n i 1

ai

n i 1 Ni

D 1, then the generic row degree of the relation module is 0 1 .

9

Solutions of SRFR and Relation Module

Simultaneous Rational Function Reconstruction

Instance: ai, ui with deg(ui) < deg(ai), and Ni ∈ Nn, D ∈ N Solution: (vi, d) such that

• vi ≡ dui mod ai ⇐

⇒ (v, d) ∈ A, where A := {p | pR ≡ 0 mod M}

• 

  deg(vi) < Ni deg(d) < D ⇐ ⇒ rdegs(vi, d) < 0, where s = (−N1, . . . , −Nn, −D),

Theorem, [Guerrini, Lebreton, Z.]

If n

i=1 deg(ai) = n i=1 Ni + D − 1, then

the generic row degree of the relation module is (0, . . . , 0, −1).

9

Technical results

Pivot degrees and free families of monomials

• A be the relation module of a general matrix M ∈ K[x]m×n, m ≥ n
• δM ∈ Zm the pivot degrees,
• monomials in K[x]m are xiεj = (0, . . . , xi, . . . , 0),
• FδM := {xiεj}i<δM,j ordered w.r.t. s − TOP

Proposition

M is the smallest subfamily of monomials such that

mM

m

M are linearly independent

pivot degrees smallest subfamily of linearly independent monomials

10

Pivot degrees and free families of monomials

• A be the relation module of a general matrix M ∈ K[x]m×n, m ≥ n
• δM ∈ Zm the pivot degrees,
• monomials in K[x]m are xiεj = (0, . . . , xi, . . . , 0),
• FδM := {xiεj}i<δM,j ordered w.r.t. s − TOP

s-TOP order

s = (0, 1, 2) Mon (1, 0, 0) ≤ (x, 0, 0) ≤ (0, 1, 0) ≤ (x2, 0, 0) ≤ (0, x, 0) ≤ (0, 0, 1) rdegs 1 2

Proposition

M is the smallest subfamily of monomials such that

mM

m

M are linearly independent

pivot degrees smallest subfamily of linearly independent monomials

10

Pivot degrees and free families of monomials

• A be the relation module of a general matrix M ∈ K[x]m×n, m ≥ n
• δM ∈ Zm the pivot degrees,
• monomials in K[x]m are xiεj = (0, . . . , xi, . . . , 0),
• FδM := {xiεj}i<δM,j ordered w.r.t. s − TOP

Proposition

FδM is the smallest subfamily of monomials such that {mM mod M}m∈FδM are linearly independent pivot degrees ← → smallest subfamily of linearly independent monomials

10

Condition on free families of monomials

Recall that M = (0, . . . , ai, . . . , 0)i and Fd = {xiεj}i<dj.

Theorem [Villard ’97], [Pernet, Storjohann ’07]

Let d ∈ Nm. Then,

∃M s.t. {mM mod M}m∈Fd are lin. indep. ⇐ ⇒          d1 ≤ deg(a1) d1 + d2 ≤ deg(a1) + deg(a2) . . .

dmin d

dl al M s.t. mM

m

dmin are lin. indep.

Proposition

dmin

M

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Condition on free families of monomials

Recall that M = (0, . . . , ai, . . . , 0)i and Fd = {xiεj}i<dj.

Theorem [Villard ’97], [Pernet, Storjohann ’07]

Let d ∈ Nm. Then,

∃M s.t. {mM mod M}m∈Fd are lin. indep. ⇐ ⇒          d1 ≤ deg(a1) d1 + d2 ≤ deg(a1) + deg(a2) . . .

Fdmin := min{Fd | dl ≤ deg(al)}

• ∃M′ s.t. {mM′ mod M}m∈Fdmin are lin. indep.

Proposition

dmin

M

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Condition on free families of monomials

Recall that M = (0, . . . , ai, . . . , 0)i and Fd = {xiεj}i<dj.

Theorem [Villard ’97], [Pernet, Storjohann ’07]

Let d ∈ Nm. Then,

∃M s.t. {mM mod M}m∈Fd are lin. indep. ⇐ ⇒          d1 ≤ deg(a1) d1 + d2 ≤ deg(a1) + deg(a2) . . .

Fdmin := min{Fd | dl ≤ deg(al)}

• ∃M′ s.t. {mM′ mod M}m∈Fdmin are lin. indep.

Proposition

Fdmin ≤ FδM′

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Genericity

Proposition

If M is generic, then Fdmin = FδM and in particular dmin = δM

Idea of the proof

• Fdmin minimal ⇐

⇒ ∃M′ s.t. Fdmin = FδM′

• dmin

M

mM

m

dmin lin.indep. (generic condition on M)

For “balanced” shifts generic row degree is s p 1 p 1 p p

12

Genericity

Proposition

If M is generic, then Fdmin = FδM and in particular dmin = δM

Idea of the proof

• Fdmin minimal ⇐

⇒ ∃M′ s.t. Fdmin = FδM′

• Fdmin = FδM ⇐

⇒ {mM mod M}m∈Fdmin lin.indep. (generic condition on M)

For “balanced” shifts generic row degree is s p 1 p 1 p p

12

Genericity

Proposition

If M is generic, then Fdmin = FδM and in particular dmin = δM

Idea of the proof

• Fdmin minimal ⇐

⇒ ∃M′ s.t. Fdmin = FδM′

• Fdmin = FδM ⇐

⇒ {mM mod M}m∈Fdmin lin.indep. (generic condition on M)

For “balanced” shifts ⇒ generic row degree is ρ = δ + s = (p + 1, . . . , p + 1, p, . . . , p)

12

Summary of technical results

General case

• 1. pivot degrees δ ←

→ smallest subfamily of lin. indep. monomials Fδ,

• 2. characterization on free families of monomials Fd (Theorem),
• 3. take the minimum Fd′,
• 4. generically Fδ = Fd′ and so d′ = δ,
• 5. “balanced” shifts → ρ = δ + s = (p + 1, . . . , p + 1, p, . . . , p).

Special case SRFR

• shift s

N1 Nn D

• The construction of the proof of 2 provides a matrix of the form R,
• “balanced” shift

s 1 uniqueness. 13

Summary of technical results

General case

• 1. pivot degrees δ ←

→ smallest subfamily of lin. indep. monomials Fδ,

• 2. characterization on free families of monomials Fd (Theorem),
• 3. take the minimum Fd′,
• 4. generically Fδ = Fd′ and so d′ = δ,
• 5. “balanced” shifts → ρ = δ + s = (p + 1, . . . , p + 1, p, . . . , p).

Special case SRFR

• shift s = (−N1, . . . , −Nn, D)
• The construction of the proof of 2 provides a matrix of the form R,
• “balanced” shift ⇒ ρ = δ + s = (0, . . . , 0, −1) ⇒ uniqueness.

13

Conclusions & open problems

Theorem [Guerrini, Lebreton, Z.]

If deg(ai) = Ni + D − 1, for almost all u ⇒ uniqueness of SRFR.

Polynomial Linear System

• RFR, bigger bound

uniqueness,

• same bound, special cases

uniqueness [Olesh, Storjohann ’07] ,

• Theorem [Guerrini, Lebreton, Z.]
• open problem

same bound , for almost all v d uniqueness? Polynomial Linear System with Errors

• RFR, bigger bound

uniqueness [Boyer, Kaltofen ’14] [Kaltofen, Pernet, Storjohann ’17],

• intermediate bound for almost all errors

uniqueness [Guerrini, Lebreton, Z. ’19],

• open problem

same bound for almost all errors and for almost all v d uniqueness? 14

Conclusions & open problems

Theorem [Guerrini, Lebreton, Z.]

If deg(ai) = Ni + D − 1, for almost all u ⇒ uniqueness of SRFR.

Polynomial Linear System

• RFR, bigger bound ⇒ uniqueness,
• same bound, special cases ⇒ uniqueness [Olesh, Storjohann ’07] ,
• Theorem [Guerrini, Lebreton, Z.]
• open problem

same bound , for almost all (v, d) ⇒ uniqueness? Polynomial Linear System with Errors

• RFR, bigger bound

uniqueness [Boyer, Kaltofen ’14] [Kaltofen, Pernet, Storjohann ’17],

• intermediate bound for almost all errors

uniqueness [Guerrini, Lebreton, Z. ’19],

• open problem

same bound for almost all errors and for almost all v d uniqueness? 14

Conclusions & open problems

Theorem [Guerrini, Lebreton, Z.]

If deg(ai) = Ni + D − 1, for almost all u ⇒ uniqueness of SRFR.

Polynomial Linear System

• RFR, bigger bound ⇒ uniqueness,
• same bound, special cases ⇒ uniqueness [Olesh, Storjohann ’07] ,
• Theorem [Guerrini, Lebreton, Z.]
• open problem

same bound , for almost all (v, d) ⇒ uniqueness? Polynomial Linear System with Errors

• RFR, bigger bound ⇒ uniqueness [Boyer, Kaltofen ’14] [Kaltofen, Pernet, Storjohann ’17],
• intermediate bound for almost all errors → uniqueness [Guerrini, Lebreton, Z. ’19],
• open problem

same bound for almost all errors and for almost all (v, d) ⇒ uniqueness? 14

Thank you!

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