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)
pv min pd min for p Rational Function Reconstruction equations x . unique solution: v d 1 D N a a non trivial solution 1 unknowns Consider the homogeneous linear system Rational Function Reconstruction a du v v • 1 Instance: a , u ∈ K [ x ] with deg( u ) < deg( a ) , and N , D ∈ N Solution: ( v , d ) ∈ K [ x ] 2 such that d ≡ u mod a • deg( v ) < N , • deg( d ) < D .
pv min pd min for p Rational Function Reconstruction equations x . unique solution: v d 1 D N a a non trivial solution 1 unknowns Consider the homogeneous linear system Rational Function Reconstruction a du v v • 1 Instance: a , u ∈ K [ x ] with deg( u ) < deg( a ) , and N , D ∈ N Solution: ( v , d ) ∈ K [ x ] 2 such that d ≡ u mod a ⇒ v ≡ du mod a • deg( v ) < N , • deg( d ) < D .
pv min pd min for p Rational Function Reconstruction unknowns x . unique solution: v d 1 D N a a non trivial solution 1 equations Rational Function Reconstruction Consider the homogeneous linear system RFR is 1 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 . • Padé Approximation : a = x f , • Cauchy interpolation : a = � f i = 1 ( x − α i )
pv min pd min for p Rational Function Reconstruction Consider the homogeneous linear system x . unique solution: v d Rational Function Reconstruction 1 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 . # equations = # unknowns − 1 = ⇒ ∃ a non trivial solution � �� � ↓ ↓ deg( a ) N + D − 1
Rational Function Reconstruction Consider the homogeneous linear system Rational Function Reconstruction 1 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 . # equations = # unknowns − 1 = ⇒ ∃ a non trivial solution � �� � ↓ ↓ deg( a ) N + D − 1 unique solution: ( v , d ) = ( pv min , pd min ) for p ∈ K [ x ] .
Vector Rational function Reconstruction Apply RFR component-wise a i Vector Rational Function Reconstruction 2 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i , D i ∈ N n Solution: ( v i , d i ) such that • v i ≡ d i u i mod a i , • deg( v i ) < N i , • deg( d i ) < D i . # equations = # unknowns − 1 = ⇒ ∃ a non trivial solution � �� � ↓ ↓ N i + D i − 1 unique solution: ( v i , d i ) = ( pv min , i , pd min , i ) for p ∈ K [ x ] .
Simultaneous Rational function Reconstruction 1 existence, uniqueness? existence, uniqueness SRFR 1 D N i a i n D Simultaneous Rational Function Reconstruction N i a i RFR 3 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that • v i ≡ d u i mod a i , • deg( v i ) < N i , • deg( d ) < D .
Simultaneous Rational function Reconstruction RFR existence, uniqueness? existence, uniqueness SRFR 1 D N i a i 1 n D N i a i 3 Simultaneous Rational Function Reconstruction Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that • v i ≡ d u i mod a i , • deg( v i ) < N i , • deg( d ) < D . Use the common denominator property, # equations = # unknowns − 1 → existence, not uniqueness ↓ ↓ � deg( a i ) � ( N i ) + D − 1
Simultaneous Rational function Reconstruction Simultaneous Rational Function Reconstruction RFR SRFR existence, uniqueness existence, uniqueness? 3 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that • v i ≡ d u i mod a i , • deg( v i ) < N i , • deg( d ) < D . � deg( a i ) = � ( N i ) + n ( D − 1 ) ≤ � deg( a i ) = � ( N i ) + D − 1
Our Result Simultaneous Rational Function Reconstruction Theorem, [Guerrini, Lebreton, Z.] 4 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that • v i ≡ du i mod a i , • deg( v i ) < N i , • deg( d ) < D . If � n i = 1 deg( a i ) = � n i = 1 N i + D − 1 , then for almost all instances u ⇒ uniqueness.
SRFR Applications
SRFR Applications SRFR APPLICATIONS Polynomial Linear System Decoding Interleaved Reed-Solomon Codes 5 reconstruct y = A − 1 b ∈ K ( x ) n × 1 given evaluations recovering f ∈ K [ x ] n × 1 given evaluations, some erroneous Cramer’s rule → common denominator error locator polynomial → common denominator uniqueness → unique reconstruction uniqueness → unique decoding less points → lower complexity less points → more errors
is a solution, j err x j e rror locator polynomial Decoding IRS codes f 6 f ( 𝝱 1 ) f ( 𝝱 3 ) … f ( 𝝱 n ) f (x) received matrix deg( f )<k Decoding IRS ← → SRFR Instance: the received matrix ( u i , j ) of an IRS ( k , n ) → interpolators U i ( x ) Solution: ( v , d ) such that • v i ≡ dU i mod � n j = 1 ( x − α j ) ⇐ ⇒ v i ( α j ) = d ( α j ) u i , j , • deg( d ) < e + 1, • deg( v i ) < k + e + 1 .
Decoding IRS codes 6 f ( 𝝱 1 ) f ( 𝝱 3 ) … f ( 𝝱 n ) f (x) received matrix deg( f )<k Decoding IRS ← → SRFR Instance: the received matrix ( u i , j ) of an IRS ( k , n ) → interpolators U i ( x ) Solution: ( v , d ) such that • Λ f i ≡ Λ U i mod � n j = 1 ( x − α j ) ⇐ ⇒ Λ( α j ) f ( α j ) = Λ( α j ) u i , j , • deg( d ) < e + 1, • deg( v i ) < k + e + 1 . (Λ f , Λ) is a solution, Λ = � j err ( x − α j ) e rror locator polynomial
Solutions of SRFR and Relation Module R row degrees What about the degree constraints? D d • N i v i • Simultaneous Rational Function Reconstruction 7 Id Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that � � ⇐ ⇒ ( v , d ) ≡ 0 mod � ( 0 , . . . , a i , . . . , 0 ) � i − u � �� � • v i ≡ du i mod a i M � �� � ⇐ ⇒ ( v , d ) ∈ A , where A := { p | p R ≡ 0 mod M}
Solutions of SRFR and Relation Module R row degrees What about the degree constraints? D d N i v i Simultaneous Rational Function Reconstruction 7 Id Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that � � ⇐ ⇒ ( v , d ) ≡ 0 mod � ( 0 , . . . , a i , . . . , 0 ) � i − u � �� � • v i ≡ du i mod a i M � �� � ⇐ ⇒ ( v , d ) ∈ A , where A := { p | p R ≡ 0 mod M} • deg( v i ) < N i , • deg( d ) < D
Solutions of SRFR and Relation Module Id row degrees What about the degree constraints? Simultaneous Rational Function Reconstruction R 7 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that � � ⇐ ⇒ ( v , d ) ≡ 0 mod � ( 0 , . . . , a i , . . . , 0 ) � i − u � �� � • v i ≡ du i mod a i M � �� � ⇐ ⇒ ( v , d ) ∈ A , where A := { p | p R ≡ 0 mod M} • deg( v i ) < N i , • deg( d ) < D
Solutions of SRFR and Relation Module Id Simultaneous Rational Function Reconstruction R 7 Instance: a i , u i with deg( u i ) < deg( a i ) , and N i ∈ N n , D ∈ N Solution: ( v i , d ) such that � � ⇐ ⇒ ( v , d ) ≡ 0 mod � ( 0 , . . . , a i , . . . , 0 ) � i − u � �� � • v i ≡ du i mod a i M � �� � ⇐ ⇒ ( v , d ) ∈ A , where A := { p | p R ≡ 0 mod M} • deg( v i ) < N i , • deg( d ) < D What about the degree constraints? → row degrees
rdeg s p s p i s p 2 3 0 2 1 2 1 3 0 4 1 3 1 2 0 5 1 7 1 x 3 1 3 x 4 1 2 Example: P Defjnition: The pivots are on the diagonal. Ordered Weak Popov form pivot degree 2 x x Row degrees and Ordered Weak Popov basis 2 x 3 4 x 3 2 x x 2 x 2 2 x 4 x 4 3 x 5 x 6 1 Ordered weak Popov Basis row degrees of the module uniquely defjned 1 2 p 1 4 p 2 4 4 deg s 2 2 x 2 1 s 1 2 3 deg 2 x 3 5 3 4 deg s 1 2 1 3 x s p i rdeg p 1 2 3 deg 2 8 x 3 + 2 2 x 2 + 5 3 x + 2 → rdeg ( p ) = max(deg( p i )) = 3 → rdeg s ( p ) = max(deg s ( p i )) = 4
3 0 2 1 2 1 3 0 4 1 3 1 2 0 5 1 7 1 x 2 2 x x 3 Row degrees and Ordered Weak Popov basis x 2 4 2 x 3 1 1 2 x 4 x 4 3 x 5 x 6 x Ordered weak Popov Basis row degrees of the module uniquely defjned 2 x 1 p 4 deg 3 2 1 s 1 2 1 3 x deg s 4 2 Ordered Weak Popov form Defjnition: The pivots are on the diagonal. Example: P x 3 1 8 x 3 + 2 2 x 2 + 5 3 x + 2 → rdeg ( p ) = max(deg( p i )) = 3 → rdeg s ( p ) = max(deg s ( p i )) = 4 pivot degree = deg s ( p 2 ) = 2
Recommend
More recommend
