Computational Algebra: Big Ideas Ioannis Z. Emiris Dept. of - PowerPoint PPT Presentation

Integer Determinant Given is integer matrix [ a ij ], max entry length L = max ij { lg | a ij |} : Worst-case optimal bound on value [Hadamard] : n � a i � 2 ≤ n n/ 2 max {| a ij |} n . � | det A | ≤ i =1 O ∗ ( nL ) evalua- 1. Chinese remaindering avoids intermediate swell: tions modulo constant-length primes, each in O ∗ ( n 2 . 38 ); Lagrange in O ∗ B ( n 2 L 2 ). Total: O ∗ B ( n 3 . 38 L ) . 2. Avoid rationals [Bareiss’68] in � n i =1 n 2 iL = O ∗ B ( n 4 L ). Let [12 k ] = | a ij : i = 1 , 2 , 3 , j = 1 , 2 , k | : Multiply by a 11 rows 2 . . . , n , eliminate:     a 11 a 12 · · · a 11 a 12 · · · · · ·  → 0 a 11 a 22 − a 12 a 21 a 11 a 23 − a 13 a 21 0 a 11 a 22 − a 12 a 21 · · · · · ·    0 a 11 a 32 − a 12 a 31 a 11 a 33 − a 13 a 31 0 0 a 11 [123] a 11 [124] 3. Baby steps / giant steps O B ( n 3 . 2 L ) [Kaltofen-Villard’01]

n × n linear system rank( M ) = r ≤ n : • r = n ⇒ ∃ ! solution. • r < n ⇒ system defined by r equations. remaining equations trivial (0=0) implies ∞ roots. existence of incompatible equation (0=b) implies no roots. rank( M ) also defined for rectangular M .

Structured matrices Defined by O ( n ) elements, matrix-vector product is quasi-linear. Two important examples: Vandermonde: matrix-vector multiply and solving in O A ( n log 2 n ). • • Rectangular matrix is Toeplitz iff M ( a + i, b + i ) = M ( a, b ), i > 0, when defined, i.e. constant diagonals. Lower triangular * vector is polynomial multiplication, hence in O ∗ A ( n ); same for vector * upper triangular. ◦ More types: Hankel (constant anti-diagonals), Cauchy, Hilbert. Thm [Wiedemann (Lanszos)]. Matrix determinant reduced to O ∗ ( n ) matrix-vector products. Proof. Krylov sequence M i v computed as M ( M i − 1 v ), charpoly χ ( λ ) = det( M − λI ) = ( − 1) ± 1 λ n ± tr( M ) λ n − 1 + · · · ± det M . Caley-Hamilton thm: χ ( M ) = 0, so χ ( M ) v = 0. Berlekamp-Massey: finds k -recurrence from 2 k (vector) elements.

Toeplitz example P 1 ( x ) = x 4 − 2 x 3 + 3 x + 5 , P 2 ( x ) = 5 x 3 + 2 x − 11 . Upper triangular Toeplitz T has rows corresponding to P 2 multiples: x 4 P 2   5 0 2 − 11 0 x 3 P 2 5 0 2 − 11     x 2 P 2   5 0 2 − 11     5 0 2 − 11 xP 2     − 11 5 0 2 0 P 2 Row vector v = [1 , − 2 , 0 , 3 , 5] expresses P 1 , then Vector-matrix multi- plication vT is equivalent to polynomial multiplication ( P 1 P 2 )( x ) = 5 x 7 − 10 x 6 + 2 x 5 + 47 x 3 + 6 x 2 − 23 x − 55 . If multiplying polynomials of degree d costs F ( d ), then multiplying d × d Toeplitz matrix by vector is in O ( F ( d )) .

Real numbers

Univariate real solving

Univariate solving • Counting / Exclusion ◦ Interval arithmetic (cf. Matlab) ◦ Descartes’ rule, Bernstein basis (fast) • Sturm sequences ◦ Thom’s encoding (good asymptotics) • Approximation Numeric solvers O ( d 3 L ) ◦ ◦ Continued Fractions [E-Tsigaridas] (fast) Polynomial in Z [ x ] of degree d and bitsize L . Input size in O ( dL ), output in Ω( dL ).

Bit complexity of exact solvers Cont.Frac. Sturm Descartes Bernstein O ∗ (2 L ) O ∗ ( d 7 L 3 ) O ∗ ( d 6 L 2 ) [Uspensky48] [Heidel’71] [Collins,Akritas’76] [LaneReisenfeld81] O ( d 5 L 3 ) O ∗ ( d 6 L 3 ) O ∗ ( d 5 L 2 ) O ∗ ( d 6 L 3 ) [Akritas’80] [Davenport’88] [Krandick’95] [MourrainVrahatis] [Johnson’98] [-Yakoubson’04] O ∗ ( d 8 L 3 ) O ∗ ( d 4 L 2 ) [Sharma07] [DuSharmaYap05] [EigenwilligSharmaYap06] [E,Mourrain,T’06] + square-free + multiplicities [E,Mourrain,Tsigaridas’06] O ∗ ( d 4 L 2 ) O ∗ ( r d 2 L 2 ) O ∗ ( d 3 L 2 ) [ET’06] [E,Tsigaridas] Polynomial in Z [ x ] of degree d and bitsize L . Best numerical algorithm in O ( d 3 L ), input = O ( dL ). Worst-case vs. average-case complexities, r = #real-roots.

Sturm theory

Sturm sequences Definition. Given univariate polynomials P 0 , P 1 ∈ R [ x ], their Sturm sequence is any (pseudo-remainder) sequence of polynomials P 0 , P 1 , . . . , P n ∈ R [ x ] , n ≥ 1 such that α i P i − 1 = QP i + β i P i +1 , i = 1 , . . . , n − 1 , for some Q ∈ R [ x ] , α i , β i ∈ R , and α i β i < 0. ′ δηs Remember Eυκλǫι

Example of Sturm sequence Input: f i = α i x 2 − 2 β i x + γ i , i = 1 , 2 . Hypothesis: the f i are relatively prime, α i , ∆ i � = 0. The Sturm sequence ( P i ) of f 1 , f ′ 1 f 2 : P 0 ( x ) = f 1 ( x ) f ′ P 1 ( x ) = 1 ( x ) f 2 ( x ) − f 1 ( x ) P 2 ( x ) = 2 α 1 [ − ( α 1 K + 2 α 2 ∆ 1 ) x + ( γ 1 J − α 1 J ′ )] P 3 ( x ) = − α 1 ∆ 1 ( α 1 K + 2 α 2 ∆ 1 ) 2 ( G 2 − 4 JJ ′ ) P 4 ( x ) = = − α 1 ∆ 1 ( α 1 G − 2 β 1 J ) 2 ( G 2 − 4 JJ ′ )

Root counting Theorem [Tarski]. Suppose that ◦ f 0 , f 1 ∈ R [ x ] are relatively prime, ◦ f 0 is square-free, and ◦ p < q are not roots of f 0 . Then, for any Sturm sequence P = ( f 0 , f ′ 0 f 1 , . . . ), � V P ( p ) − V P ( q ) = sign( f 1 ( ρ )) , f 0 ( ρ )=0 ,p<ρ<q where V P ( p ) := #sign variations in P 0 ( p ) , . . . , P n ( p ). The Sturm sequence here may be ( f 0 , f ′ 0 f 1 , − f 0 , . . . ).

More uses of Sturm sequences Corollary. For p < q non-roots of f ∈ R [ x ], the number of distinct real roots of f in ( p, q ) equals V f,f ′ ( p ) − V f,f ′ ( q ). Proof. Let f 0 = f, f 1 = 1 in Tarski’s theorem. Theorem [Schwartz-Sharir]. For square-free f 0 , f 1 ∈ R [ x ] and p < q non-roots of f 0 , sign( f ′ � V f 0 ,f 1 ( p ) − V f 0 ,f 1 ( q ) = 0 ( ρ ) f 1 ( ρ )) . f 0 ( ρ )=0 ,p<ρ<q Yields previous theorem by using f 0 , f ′ • 0 f 1 . [Yap: Fundamental Problems of Algorithmic Algebra, 2000]

Generalizations of Sturm theory

Systems of univariate polynomials Recall [Tarski]. For f 0 , f 1 ∈ R [ x ] relatively prime, f 0 square-free and p < q not roots of f 0 , consider the Sturm sequence ( f 0 , f ′ 0 f 1 , . . . ). Then � V ( p ) − V ( q ) = sign( f 1 ( ρ )) . f 0 ( ρ )=0 ,p<ρ<q This equals # { ρ ∈ ( p, q ) : f 0 ( ρ ) = 0 , f 1 ( ρ ) > 0 }− # { ρ ∈ ( p, q ) : f 0 ( ρ ) = 0 , f 1 ( ρ ) < 0 } . Algorithm [Ben-Or,Kozen,Reif], [Canny]. Compute n � # { ρ ∈ ( p, q ) : P 0 ( ρ ) = 0 , P i ( ρ ) ⊗ i 0 } , ⊗ i ∈ { <, > } . i =1

Generalized Sturm sequences Definition. Given univariate polynomials P 0 , P 1 ∈ R [ x ], where P 0 is square-free, their generalized Sturm sequence over an interval [ a, b ] ⊂ R ∪ {−∞ , + ∞} is any sequence P 0 , P 1 , . . . , P n ∈ R [ x ] , n ≥ 1 s.t. 1. P 0 ( a ) P 0 ( b ) � = 0, 2. ∀ c ∈ [ a, b ] , P n ( c ) � = 0, 3. ∀ c ∈ [ a, b ] , P j ( c ) = 0 ⇒ P j − 1 ( c ) P j +1 ( c ) < 0, 4. ∀ c ∈ [ a, b ] : P 0 ( c ) = 0 ⇒ ∃ [ c 1 , c ) , ( c, c 2 ] s.t. u ∈ [ c 1 , c ) ⇒ P 0 ( u ) P 1 ( u ) < 0 and u ∈ ( c, c 2 ] ⇒ P 0 ( u ) P 1 ( u ) > 0. Corollary (Existance). For any P 0 , P 1 ∈ R [ x ], the previously-defined Sturm sequence, using the pseudo-remainders and starting with P 0 / gcd( P 0 , P ′ 0 ) and P 1 is “generalized” over an interval [ a, b ] such that P 0 ( a ) P 0 ( b ) � = 0.

Further generalization Corollary. It is possible to omit [1. P 0 ( a ) P 0 ( b ) � = 0 ] provided that, (4) is stated only in the appropriate subinterval of [ a, b ], when c = a or c = b . Corollary. Relax (4) to require that the number of roots of P 0 ( x ) is odd between any two roots of P 1 ( x ).

Real Closed fields generalize R Definition. An ordered field K contains a positive subset P ⊂ K , ie. a ∈ K − { 0 } ⇒ a ∈ P xor − a ∈ P . √ 2) ≡ Q [ x ] / � x 3 − 2 � . Examples: Q , R , Q ( ǫ ) , R ( x ) , Q ( 3 Counter-example: C . Definition. A real closed field K is • ordered (hence contains positive P ⊂ K ), ⇒ √ a ∈ P (ie. x 2 = a has a root in P ), • a ∈ P • equations of odd degree have a root in P . Examples: R , R ( ǫ ) , R ( ǫ 1 , ǫ 2 ). √ Counter-example: Q , algebraic closure Q , Q ( 3 2). Sturm sequences are defined, and all stated properties hold, for polynomials over real closed fields.

Descartes’ rule

Descartes’ rule of sign Theorem. The number of sign variations in the coefficients of a uni- variate polynomial exceeds the number of positive real roots by an even non-negative integer. Proof by induction, using Sturm sequences. Step: V [( x − a ) f ] = V [ f ]+ odd natural number. Corollary. If all roots of the univariate polynomial are nonzero and real, then the number of sign variations in the coefficient sequence gives precisely the number of positive roots. Proof by induction on the degree: the number of variations in the coefficients of f ( − x ) bounds the number of negative roots.

Notions of Algebraic geometry

Introduction

Single polynomial f ( x ) = c 0 + c 1 x + c 2 x 2 + · · · + c d x d ∈ K [ x ] . • Fundamental theorem of algebra: There are d roots in K . E.g. Q = Algebraic numbers. • Fundamental problem of real algebra: How many roots are real? • Fundamental problem of computational real algebra: Isolate all real roots of a given polynomial equation. • Fundamental problem of computational algebraic geometry: Iso- late/approximate all complex roots of a given polynomial system. • Fundamental problem of computational real algebraic geometry: Isolate all real roots of a given polynomial system.

Algebraic varieties f 1 , . . . , f m ∈ Q [ x 1 , . . . , x n ] . Defn. The polynomial system’s variety (or zero-set) is V ( f 1 , . . . , f m ) := { x ∈ C n : f 1 ( x ) = · · · = f m ( x ) = 0 } . Examples. V ( x 2 + 1) = {±√− 1 } , • • V ( Q [ x 1 , . . . , x n ]) = ∅ , V ( ∅ ) = C n . • Properties. • S ⊂ T ⇒ V ( T ) ⊂ V ( S )

Dimension of a variety Def. dim( V ) = #degrees of freedom of V = #parameters for covering V • dim(point) = 0 , dim(line) = 1 , dim(surface) = 2. dim( V ) = n ⇔ V = C n . • • dim V ( f i ) = n − 1 generically. Def. Dimension dim( V ) := max C { dim( C ) : component C ⊂ V } . • dim( V ) = 0 ⇔ V = point set (iff finite cardinality). • dim( V ) = 1 ⇔ V contains a curve (possibly straight line), may contain points, but no component of dim ≥ 2. • dim( V ) = 2 ⇔ V contains a surface (possibly planar), may contain 0-dim or 1-dim components, but no higher-dim component.

Algebraic varieties (cont’d) System f 1 , . . . , f m ∈ K [ x 1 , . . . , x n ]. • Well-constrained: m = n , generically 0-dim variety. • Over-constrained: m > n , generically no roots (empty). • Under-constrained: m < n , generically ∞ roots. Lemma. V ( f 1 , . . . , f m ) = V ( f 1 ) ∩ · · · ∩ V ( f m ) ⊂ C n . • • dim( V ∩ W ) = dim( V ) − codim ( W ), where codim( W ) = n − dim( W ); clearly, dim( V ∩ W ) = dim( W ) − codim ( V ). E.g. C 2 : V, W =curves, dim( W ∩ V ) = 0 (points). E.g. C 3 : V, W surfaces, dim( W ∩ V ) = 1 (curve). E.g. C 3 : V =surface, W =curve, dim( W ∩ V ) = 0.

n × n linear system rank( M ) = r ≤ n : • r = n ⇒ ∃ ! solution. • r < n ⇒ system defined by r equations. remaining equations trivial (0=0) implies ∞ roots. existence of incompatible equation (0=b) implies no roots.

Hilbert’s Nullstellensatz

Algebraic ideals Given a polynomial ring R = K [ x 1 , . . . , x n ], • a subring S ⊂ R is closed under addition and multiplication: a, b ∈ S ⇒ a + b, ab ∈ S ; • an (algebraic) ideal I ⊂ R is closed under addition and multiplication by any ring element: a, b ∈ I, p ∈ R ⇒ a + b, ap ∈ I . E.g. � x 2 , x 5 � = � x 2 � , � x, x + y � = � x, y � . Fact. Given a set of polynomials, all elements in the generated (alge- braic) ideal vanish at the set’s variety. Corollary. The ideal is the largest set of polynomials vanishing precisely at this variety.

Varieties vs Ideals Definition. Given set X ⊂ C n , J ( X ) := { f ∈ Q [ x ] : f ( x ) = 0 , ∀ x ∈ X } . Fact. J ( X ) is an ideal. Properties. J ( C n ) = ∅ , J ( ∅ ) = Q [ x ], • • X ⊂ Y ⇒ J ( Y ) ⊂ J ( X ), • X = V ( J ( X )) • S ⊂ J ( V ( S )): when is it tight? Counter-example: � x 2 � � = J ( { 0 } ) = � x � : ◦ How do the roots of x and x 2 differ?

Hilbert’s Nullstellensatz Recall definition J ( X ) := { f ∈ Q [ x ] : f ( x ) = 0 , ∀ x ∈ X ⊂ C n } . Defn. Given an ideal I in a commutative ring R , its radical ideal is √ I := { r ∈ R | r n ∈ I, ∃ n > 0 } . √ Property. I ⊂ I . Intuition: taking the radical removes the multiplicities. � � Eg. In ring Z : � 8 � = � 2 � , � 12 � = � 6 � , � � � x 3 � = � x � , � x 2 , x − 2 y, y 3 � = � x, y � . In a polynomial ring: √ Hilbert’s zeroes theorem. J ( V ( I )) = I. Specifies the algebra-geometry dictionary.

Polynomial Degree

Degree Defn: (total) degree of polynomial F ( x 1 , . . . , x n ) is the maximum sum of exponents in any monomial (term). E.g. deg( x 2 − xy 2 + z ) = 3. We also talk of degree in some variable(s). E.g.: deg x ( F ) = 2 , deg y ( F ) = 2 , deg z ( F ) = 1. The polynomial is homogeneous (wrt to all n variables) if all monomials have the same degree. E.g. x 2 w − xy 2 + zw 2 . Here w � = 0 is the homogenizing variable. So, for every (affine) root ( x, y, z ) ∈ C 3 there is now a (projective) root ( x : y : z : 1) ∈ P 3 .

Intersection theory Geometrically, deg f ( x 1 , . . . , x n ) equals the number of intersection points of f ( x 1 , . . . , x n ) = 0 with a generic line in C n . Defn. The degree of a variety V is #points in the intersection of V with a generic linear subspace L of dimension = codim( V ): deg V = #( V ∩ L ) : dim L = codim V. E.g. curve V ⊂ C 3 defined by f ( x, y, z ) = g ( x, y, z ). L is a generic plane.

Number of roots C or P n or P ( C ) n is the following Defn. The complex projective space P n set of equivalence classes: ( α 0 : · · · : α n ) ∈ C n +1 − { 0 n +1 } | α ∼ λα, λ ∈ C ∗ � � = = { (1 : β ) | β ∈ C n } ∪ { (0 : β ) | β ∈ C n − { 0 n } , β ∼ λβ } . E.g. n = 1: P 1 ≃ C ∪ { (0 : 1) } . Theorem [B´ ezout,1790]. Given (homogeneous) f 1 , . . . , f n ∈ K [ x 1 , . . . , x n ], the number of its common roots (counting multiplic- ities) in P ( K ) n is bounded by n � deg f i , i =1 where deg( · ) is the polynomial’s total degree. The bound is exact for generic coefficients. Note: The theorem considers dense polynomials.

Polynomial system solving

A perspective. . . on La Boca

A perspective. . . on system solving Input: n polynomial equations in n variables, coefficients in a ring (e.g. Z , R , C ). Output: All n -vectors of values s.t. all polynomials evaluate to 0.

Type Algebraic Analytic Approach Combine constraints Use values (or signs) Computation Exact (+ possibly numerical) Numerical mostly Methods Matrix-based: resultant Newton-like, optimization, symbolic-numeric computation discretization + exploit structure + simple, fast + continuity w.r.t. coefficients − local, may need initial point − high-dimensional components O ∗ b ( d n ) Gr¨ obner bases Exclusion, interval, + complete information topological degree − discontinuity w.r.t. coefficients + simple, flexible, robust b ( d n 2 ), else O ∗ b ( d 2 n ) dimension=0: O ∗ + focuses on given domain Characteristic sets − costly for large n O ∗ b ( d n ), else O ∗ b ( d n 2 ) dimension=0: Normal forms, boundary bases O ∗ b (log D ǫ ) Straight-line programs Homotopy continuation express evaluation + exploit structure − divergent paths

Resultants

Resultant definition Given n + 1 Laurent polynomials f 0 , . . . , f n ∈ K [ x 1 , . . . , x n , x − 1 1 , . . . , x − 1 n ] with indeterminate coefficients � c , their projective, resp. toric / sparse, resultant is the unique (up to sign) irreducible polynomial R ( � c ) ∈ Z [ � c ] such that R ( � c ) = 0 ⇔ ∃ ξ = ( ξ 1 , . . . , ξ n ) ∈ X : f 0 ( ξ ) = · · · = f n ( ξ ) = 0 where the variety X equals: the projective space P n over the algebraic closure K , • resp. the toric variety X, ( K ∗ ) n ⊂ X ⊂ P N . • [van der Waerden, Gelfand-Kapranov-Zelevinsky, Cox-Little-O’Shea]

Resultant degree The projective, resp. toric, resultant polynomial R ∈ Z [ � c ] is sepa- rately homogeneous in the coefficients of each f i , with degree equal to � j � = i deg f j (B´ ezout’s number), resp. the n -fold mixed volume: MV − i := MV( f 0 , . . . , f i − 1 , f i +1 , . . . , f n ) , provided the supports of the f i generate Z n . Generalizations The toric resultant reduces to: • the determinant of the coefficient matrix of a linear system, • the Sylvester or B´ ezout determinant of 2 univariate polynomials, • the projective resultant for n +1 dense polynomials, where the toric variety equals P n and MV − i = � j � = i deg f j .

Matrix formulae • Resultant matrix: The resultant divides the determinant. • Rational, Macaulay-type formula: The resultant equals the ratio of two determinants. • Determinantal (optimal) formula: the resultant equals a determi- nant • Polynomial formula: A power of the resultant equals the determi- √ nant, Pfaffian when R = det M . • Poisson formula. • Determinantal from rational formula [Kaltofen-Koiran’08] • Matrix formulae allow system solving by: an eigenproblem, u - resultant, primitive/separating element (RUR).

Resultant matrices • n = 1: B´ ezout 1779, Sylvester 1840. • B´ ezout: [Chtcherba-Kapur’00], [Kapur et.al], [Cardinal-Mourrain], [Bus´ e et al.]. • Homogeneous: Macaulay, [GKZ’94], [Jouanolou’97], [D’Andrea- Dickenstein’01], [CoxMatera08], complexes [Eisenbud-Schreyer’03]. [Canny-E’93], [E-Canny’93] ∗ , generalized [Sturmfels’94], • Toric: Jacobian [Cattani-Dickenstein-Sturmfels], [D’Andrea-E’01], com- plexes [Khetan’02], rational [D’Andrea’02], [E-Konaxis’09]. • m-homogeneous: Dixon, [GKZ], [Chionh-Goldman- Zhang98,ZG00], [Dickenstein-E’03, E-Mantzaflaris’09], [Awane- Chkiriba-Goze’05].

A bilinear example

Example: Bilinear surface A bilinear surface in R 3 is the set of values ( x 1 , x 2 , x 3 ): x i = c i 0 + c i 1 s + c i 2 t + c i 3 st, i = 1 , 2 , 3 , for s, t ∈ [0 , 1] , as well as the set of roots of some polynomial equation H ( x 1 , x 2 , x 3 ) = 0.

Modeling/CAD use parametric AND implicit/algebraic representations ⇒ need to implicitize a (hyper)surface given a (rational) parameteri- zation.

Bilinear system: Resultant matrix f i = ( c i 0 − x i ) + c i 1 s + c i 2 t + c i 3 st, i = 1 , 2 , 3 . The classical projective resultant vanishes identically. The toric (sparse) resultant has deg R = 3 · deg f i R = 6. A determinantal Sylvester-type formula for the toric resultant is: s 2 s 2 t 1 s t st   c 10 − x 1 c 11 c 12 c 13 0 0 f 1 c 20 − x 2 0 0 c 21 c 22 c 23 f 2     c 30 − x 3 0 0  c 31 c 32 c 33  f 3   R = det   0 c 10 − x 1 0 c 12 c 11 c 13 sf 1     0 c 20 − x 2 0 c 22 c 21 c 23 sf 2     0 c 30 − x 3 0 c 32 c 31 c 33 sf 3

Sparse elimination theory

Newton polytopes The support A i of a polynomial f i ∈ K [ x ± 1 1 , . . . , x ± 1 n ], s.t. c ij x a ij , � f i = c ij � = 0 , j is defined as the set A i := { a ij ∈ Z n : c ij � = 0 } . The Newton polytope Q i ⊂ R n of f i is the Convex Hull of all a ij ∈ A i . c 11 + c 12 xy + c 13 x 2 y + c 14 x f 1 = c 21 y + c 22 x 2 y 2 + c 23 x 2 y + c 24 x + c 25 xy Example: f 2 = f 3 = c 31 + c 32 y + c 33 xy + c 34 x

Mixed volume 1. The mixed volume MV( P 1 , . . . , P n ) ∈ R of convex polytopes P i ⊂ R n • is multilinear wrt Minkowski addition and scalar multiplication: MV( P 1 , . . . , λP i + µP ′ i , . . . , P n ) = = λ MV( P 1 , . . . , P i , . . . , P n )+ µ MV( P 1 , . . . , P ′ i , . . . , P n ) , λ, µ ∈ R , • st. MV( P 1 , . . . , P 1 ) = n ! vol( P 1 ). 2. Equivalently, vol( λ 1 P 1 + · · · + λ n P n ) is a polynomial in scalar variables λ 1 , . . . , λ n , with multilinear term MV( P 1 , . . . , P n ) λ 1 · · · λ n .   ( − 1) n −| I | vol  . � � 3. Exclusion-Inclusion definition: MV := Q i i ∈ I I ⊂{ 1 ,...,n }

Mixed Volume characterization MV: vtx( Q i ) ⊂ Z n Property Generic number of isolated solutions ∗ ) n | · · · = f i ( x ) = · · · = 0 } ∈ Z ≥ 0 MV( . . . , Q i , . . . ) # { x ∈ ( K Invariance by MV( . . . , Q j , . . . , Q i , . . . ) = # { x | · · · = f j ( x ) = · · · = f i ( x ) = · · · = 0 } = permutation = MV( . . . , Q i , . . . , Q j , . . . ) = # { x | · · · = f i ( x ) = · · · = f j ( x ) = · · · = 0 } MV( . . . , Q i + Q ′ # { x | · · · = ( f i f ′ Linearity wrt i , . . . ) = i )( x ) = · · · = 0 } = Minkowski = MV( . . . , Q i , . . . )+ = # { x | · · · = f i ( x ) = · · · = 0 } + +MV( . . . , Q ′ +# { x | · · · = f ′ addition i , . . . ) i ( x ) = · · · = 0 } # { x | · · · = ( f i ( x )) λ = · · · = 0 } = Linearity wrt MV( . . . , λQ i , . . . ) = scalar product = λ MV( . . . , Q i , . . . ) = λ # { x | · · · = f i ( x ) = · · · = 0 } # { x | · · · = f i ( x ) + cx a = · · · = 0 } ≥ Monotone MV( . . . , Q i ∪ { a } , . . . ) ≥ wrt volume ≥ MV( . . . , Q i , . . . ) ≥ # { x | · · · = f i ( x ) = · · · = 0 } [Kushnirenko] MV( Q 1 , . . . , Q 1 ) = n ! V ( Q 1 ) # { x | f 1 ( x ) = · · · = f n ( x ) = 0 } = n ! V ( Q 1 )

Bernstein (BKK) bound Theorem [Bernstein’75,Kushnirenko’75,Khovanskii’78] [Danilov’78] : Given polynomials f 1 , . . . , f n ∈ K [ x ± 1 1 , . . . , x ± 1 n ], for any field K , the number of common isolated zeros in ( K − { 0 } ) n , counting mul- tiplicities, is bounded by the mixed volume of the Newton polytopes MV( Q 1 , . . . , Q n ) (irrespective of the variety’s dimension). Dense homogeneous: MV( Q 1 , . . . , Q n ) = � n i =1 d i = B´ ezout’s bound, where d i = deg( f i ) and Q i = simplex { 0 , ( d i , 0 , . . . , 0) , . . . , (0 , . . . , 0 , d i ) } . Dense multi-homogeneous: MV( Q 1 , . . . , Q n ) = m-B´ ezout’s bound: r n y n j � � the coefficient of in ( d i 1 y 1 + · · · + d ir y r ) , j j =1 i =1 where deg X j f i = d ij , j = 1 , . . . , r , and X j contains n j variables.

Example: mixed subdivision for well-constrained problem Given f 1 = c 11 + c 12 xy + c 13 x 2 y + c 14 x , f 3 = c 31 + c 32 y + c 33 xy + c 34 x , construct their Newton polytopes in R 2 • • compute a mixed subdivision of the Minkowski Sum (3 mixed cells) • compute the Mixed Volume using the formula MV= � σ V ( σ ), over all mixed cells σ of the mixed subdivision (here MV=3).

Resultant definition Given n + 1 Laurent polynomials f 0 , . . . , f n ∈ K [ x 1 , . . . , x n , x − 1 1 , . . . , x − 1 n ] with indeterminate coefficients � c , their projective, resp. toric / sparse, resultant is the unique (up to sign) irreducible polynomial R ( � c ) ∈ Z [ � c ] such that R ( � c ) = 0 ⇔ ∃ ξ = ( ξ 1 , . . . , ξ n ) ∈ X : f 0 ( ξ ) = · · · = f n ( ξ ) = 0 where the variety X equals: the projective space P n over the algebraic closure K , • resp. the toric variety X, ( K ∗ ) n ⊂ X ⊂ P N . • [van der Waerden, Gelfand-Kapranov-Zelevinsky, Cox-Little-O’Shea]

Resultant degree The projective, resp. toric, resultant polynomial R ∈ Z [ � c ] is sepa- rately homogeneous in the coefficients of each f i , with degree equal to � j � = i deg f j (B´ ezout’s number), resp. the n -fold mixed volume: MV − i := MV( f 0 , . . . , f i − 1 , f i +1 , . . . , f n ) , provided the supports of the f i generate Z n . Generalizations The toric resultant reduces to: • the determinant of the coefficient matrix of a linear system, • the Sylvester or B´ ezout determinant of 2 univariate polynomials, • the projective resultant for n +1 dense polynomials, where the toric variety equals P n and MV − i = � j � = i deg f j .

Lifting in the Sylvester case f 0 = c 00 + c 01 x, f 1 = c 10 + c 11 x + c 12 x 2 RC(2) = (1; 2) ie. x 2 �→ x 2 − 2 f 1 .

Mixed subdivision of a linear system RC(1 , 2) = [2 , (0 , 1)] ie. x 1 x 2 2 �→ x (1 , 2) − (0 , 1) f 2 = x (1 , 1) f 2 RC(1 , 1) = [1 , (0 , 0)] ie. x 1 x 2 �→ x (1 , 1) − (0 , 0) f 1 = x (1 , 1) f 1 RC(2 , 1) = [0 , (1 , 0)] ie. x 2 1 x 2 �→ x (2 , 1) − (1 , 0) f 0 = x (1 , 1) f 0 x 2 x 1 x 2 1 x 2 x 1 x 2 2   c 01 c 02 c 03 x 1 x 2 f 0 M = c 11 c 12 c 13 x 1 x 2 f 1     c 21 c 22 c 23 x 1 x 2 f 2

Example: subdivision-based matrix c 11 + c 12 xy + c 13 x 2 y + c 14 x, = f 1 c 21 y + c 22 x 2 y 2 + c 23 x 2 y + c 24 x, f 2 = f 3 = c 31 + c 32 y + c 33 xy + c 34 x. 1 , 0 2 , 0 0 , 1 1 , 1 2 , 1 3 , 1 0 , 2 1 , 2 2 , 2 3 , 2 4 , 2 1 , 3 2 , 3 3 , 3 4  (1 , 0) x c 11 c 14 0 0 c 12 c 13 0 0 0 0 0 0 0 0 (2 , 0) x c 31 c 34 0 c 32 c 33 0 0 0 0 0 0 0 0 0   (0 , 1) y 0 0 c 11 c 14 0 0 0 c 12 c 13 0 0 0 0 0  (1 , 1) xy 0 0 0 c 11 c 14 0 0 0 c 12 c 13 0 0 0 0   (2 , 1) c 24 0 c 21 0 c 23 0 0 0 c 22 0 0 0 0 0   (3 , 1) x 0 c 24 0 c 21 0 c 23 0 0 0 c 22 0 0 0 0   (0 , 2) y 0 0 c 31 c 34 0 0 c 32 c 33 0 0 0 0 0 0   (1 , 2) xy 0 0 0 c 31 c 34 0 0 c 32 c 33 0 0 0 0 0  (2 , 2) x 2 y 2  0 0 0 0 0 0 0 0 c 11 c 14 0 0 0 c 12  (3 , 2) x 2 y 0 0 0 0 0 0 0 0 0 0  c 31 c 34 c 32 c 33  (4 , 2) x 2 y 0 0 0 0 0 0 0 0 0 0 0 c 24 c 21 c 23   (1 , 3) xy 2 0 0 0 0 0 0 0 0 0 0 c 31 c 34 c 32 c 33   (2 , 3) y 0 0 0 c 24 0 0 c 21 0 c 23 0 0 0 c 22 0  (3 , 3) x 2 y 2  0 0 0 0 0 0 0 0 c 31 c 34 0 0 c 32 c 33  (4 , 3) x 3 y 2 0 0 0 0 0 0 0 0 0 c 31 c 34 0 0 c 32 dim M = 15, greedy [Canny-Pedersen]: 14, incremental [E-Canny]: 12. Mixed volumes = 4 , 3 , 4 ⇒ deg R tor = 11 while deg(classical resultant) = 26.

Polynomials of arbitrary support

Matrices of Sylvester-type Algorithms: subdivision-based [Canny-E’93,’00] , incremental [E-Canny’95] yield a square matrix M of the sparse/toric resultant, such that: det( M ) �≡ 0 , R | det( M ) , deg f 0 det( M ) = deg f 0 R, where R is the toric resultant. R = det( M ) / det( M ′ ), Rational form [D’Andrea’02] : where M ′ is a submatrix of M , generalizing Macaulay’s construction.

Matrix construction [Canny,E’93,00] 1. Pick (affine) liftings ω i : Z n → R : supp( f i ) → Q . 2. Define (tight coherent polyhedral) mixed subdivision of the Minkowski sum Q = Q 0 + · · · + Q n of the Newton polytopes. Maximal cells are uniquely expressed as σ = F 0 + · · · + F n , with dim F 0 + · · · + dim F n = n, where F i is a face of Q i . σ is i − mixed ⇐ ⇒ ∃ ! i : dim F i = 0. 3. For every point p ∈ E = ( Q + δ ) ∩ Z n , ∃ unique σ + δ ∋ p . Define function RC( p ) = ( i, F i ) : unique if σ i -mixed, else pick max i . 4. Construct resultant matrix M with rows/columns indexed by E : for p, q ∈ E , element ( p, q ) is the coefficient of x q in x p − a i f i : p − δ ∈ σ = F 0 + · · · + a i + · · · + F n (max i ), i.e. RC( p ) = ( i, a i ).

Correctness Lemma. RC( p ) = ( i, a i ) ⇒ support( x p − a i f i ) ⊂ E . Proof. p ∈ σ + δ ⊂ Q 0 + · · · + Q i − 1 + a i + Q i +1 + · · · + Q n + δ implies p − a i ∈ � i � = j Q i + δ , hence p − a i + q ⊂ E for all q ∈ supp( f i ). Corollary. The diagonal entry at the row indexed by p contains the f i coefficient of x a i . Proof. Consider the row indexed by p , s.t. RC( p ) = ( i, a i ). Then, the f i coefficient of x a i is the coefficient of x p in x p − a i f i , hence it appears at the column indexed by p .

Incremental algorithm [E-Canny’95] Idea: The rows express x b f i : b ∈ Q − i ∩ Z n , where Q − i = Q 0 + · · · + Q i − 1 + Q i +1 + · · · + Q n so that column monomials ⊂ � i Q i . 1. Sort Q − i ∩ Z n on their distance dist v ( · ) from the boundary of Q − i along some vector v ∈ Q n . 2. Define the rows of M by points B i = { b : dist v ( b ) > β } , for bound β ∈ R . The columns are indexed by ∪ i ∪ b ∈ B i supp( x b f i ). 3. Enlarge M by decreasing β until M (i) has at least as many rows as columns and (ii) is generically of full rank. For multihomogeneous systems: Deterministic vector v yields: • exact matrices if possible [Sturmfels-Zelevinsky’94] , • otherwise minimum matrices [Dickenstein-E’02] . Complexity in ∼ e 2 n (deg R ) 2 (by quasi-Toeplitz structure)

Unmixed multihomogeneous systems Partition the variables to r subsets: every polynomial is homogeneous in each subset. The i -th subset has l i + 1 homogeneous variables, of total degree d i . Then the polynomial is of type ( l 1 , . . . , l r ; d 1 , . . . , d r ) . Type (2 , 1; 2 , 1) : ( x 1 , x 2 , y 1 ) ∈ P 2 × P 1 : c 0 + c 1 x 1 + c 2 x 2 + c 3 x 1 x 2 + c 4 x 2 1 + c 5 x 2 2 + c 6 y 1 + c 7 x 1 y 1 + c 8 x 2 y 1 + c 9 x 1 x 2 y 1 + c 10 x 2 1 y 1 + c 11 x 2 2 y 1 . A system is of type ( l, d ) iff all polynomials are of type ( l, d ). [Sturmfels,Zelevinsky’94]. If l i = 1 or d i = 1 , ∀ i, then ∃ determinantal resultant formula i.e. det M = R . Type (2 , 1; 1 , 1): c 0 + c 1 x 1 + c 2 x 2 + c 3 y 1 + c 4 x 1 y 1 + c 5 x 2 y 1 . [Dickenstein,E’02] find minimum (non-optimal) Sylvester-type matrix; extended by [E-Mantzaflaris] The incremental algorithm [E,Canny’95] constructs all these matrices.

Rational form Recursive lifting on n , using the subdivision algorithm [D’Andrea’01]. Bilinear: f i = a i + b i x 1 + c i x 2 + d i x 1 x 2 , i = 0 , 1 , 2. Linear lift ( −∞ , . . . ) , (0 , 1 , 1 , 2) , (0 , 0 , 7 , 7), δ = ( 2 3 , 1 2 ) ⇒ dim M = 16 (numerator): a 1 b 1 0 c 1 d 1 0 0 0 0 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0 0 0 0 a 0 b 0 c 0 d 0   0 0 0 0 0 0 0 0 0 0 0 0 a 1 b 1 c 1 d 1     0 0 0 0 0 0 0 0 0 0 0 0 a 2 b 2 c 2 d 2     0 0 0 0 0 0 0 0 0 0 0 0 a 2 b 2 c 2 d 2     0 0 0 0 0 0 0 0 0 0 0 0 a 0 c 0 d 0 b 0   0 0 0 0 0 0 0 0 0 0 0 0  a 2 b 2 c 2 d 2    0 0 0 0 0 0 0 0 0 0 0 0 a 2 b 2 c 2 d 2   M =   0 0 0 0 0 0 0 0 0 0 0 0 a 1 b 1 c 1 d 1     0 0 0 0 0 a 1 0 0 c 1 d 1 b 1 0 0 0 0 0     0 0 0 0 0 a 2 0 0 c 2 d 2 b 2 0 0 0 0 0     0 0 a 2 0 0 c 2 0 0 0 0 d 2 b 2 0 0 0 0     0 0 0 0 0 0 a 2 b 2 0 0 0 0 c 2 d 2 0 0   0 0 0 0 0 0 0 a 2 b 2 0 0 0 0 c 2 d 2 0     0 0 0 0 0 0 0 0 a 2 b 2 0 0 0 0 c 2 d 2   0 0 0 0 0 0 0 0 a 1 b 1 0 0 0 0 c 1 d 1

Rational form: denominator a 1 0 c 1 d 1 0 0 0 0 0 0   0 b 1 0 c 1 0 0 0 0 0 0   a 2 0 c 2 d 2 0 0 0 0 0 0     0 b 2 0 c 2 0 0 0 0 0 0     0 0 a 2 b 2 c 2 d 2 0 0 0 0 M ′ =     0 0 0 0 0 0 0 a 2 c 2 d 2    0 0 0 0 0 0 0  a 1 c 1 d 1   0 0 0 0 0 0 0 0 c 2 b 2     0 0 0 0 0 0 a 2 b 2 c 2 d 2   0 0 0 0 0 0 0 0 a 2 c 2 det( M ) = ± R · det( M ′ ): M ′ is a submatrix of M , | M ′ | = − c 3 2 ( − c 1 a 2 + a 1 c 2 ) b 2 ( c 1 d 2 − d 1 c 2 )( − b 2 c 1 + b 1 c 2 ) Main step: lifting of some b ∈ Q 0 is very negative. The mixed subdivision provides all info. Open: ∃ single lifting yielding both numerator and denominator? YES if n = 2, unmixed system, or sufficiently different Newton poly- topes [E-Konaxis’11]

B´ ezout matrices

The Bezoutian Definition. For f 0 , . . . , f n ∈ K [ x 1 , . . . , x n ], the Bezoutian polynomial is   f 0 ( x ) θ 1 ( f 0 )( x, z ) · · · θ n ( f 0 )( x, z ) . . . . . . . . Θ f i ( x, z ) = det . . . .  ,    f n ( x ) θ 1 ( f n )( x, z ) · · · θ n ( f n )( x, z ) θ i ( f j )( x, z ) = f j ( z 1 , . . . , z i − 1 , x i , . . . , x n ) − f j ( z 1 , . . . , z i , x i +1 , . . . , x n ) . x i − z i θ ab x a z b , θ a,b ∈ K, a, b ∈ N n . � Let Θ f 0 ,...,f n ( x, z ) = a,b Then the Bezoutian matrix of f 0 , . . . , f n is the matrix [ θ ab ] a,b . Theorem. [Cardinal-Mourrain’96] The resultant divides all maximal nonzero minors of the Bezoutian matrix. The dimension of the matrix is O ( e n d n ) , d = max { deg f i } .

Polynomial system solving

Polynomial System Solving I Given f 1 , . . . , f n ∈ K [ x ± 1 1 , . . . , x ± 1 n ] defining a 0-dimensional radical ideal. Add polynomial f 0 = u + r 1 x 1 + · · · + r n x n , random r i , indeterminate u . Construct resultant matrix M ( u ) for f 0 , f 1 , . . . , f n . At root α , u = − � r i α i , . . . .    .     .  . . . α p α a f i ( α ) M 11 M 12 0                 . . . . . .      .    . = = . .            α q      α b f 0 ( u, α ) M 21 M 22 ( u ) 0         . . .         . . . . . . If det M 11 � = 0, let M ′ ( u ) = M 22 ( u ) − M 21 M − 1 11 M 12 , ( M ′ + uI ) v ′ dim M ′ = MV( f 1 , . . . , f n ) . α = 0 , Ratios of the entries of eigenvectors v ′ α yield α , if the q span Z n . •

Otherwise, use some entries of v α = − M − 1 11 M 12 v ′ α , where ( v α , v ′ α ) T • is the respective eigenvector of M .

Polynomial System Solving I (factoring) For f 0 = u 0 + u 1 x 1 + · · · + u n x n , with indeterminates u i , the Poisson formula implies ( u 0 + α 1 u 1 + · · · + α n u n ) m α , � R ( u 0 , . . . , u n ) = C α ∈ V ( f 1 ,...,f n ) over all roots α with multiplicity m α , where C depends on the coeffi- cients of f 1 , . . . , f n . Setting u i = r i , i = 1 , . . . , n, for random r i , we have ( u 0 + r 1 α 1 + · · · + r n α n ) m α . � R ( u 0 ) = C α Solving R ( u 0 ) for u 0 yields u 0 = − � i r i α i for all α . R ( u 0 ) is used in the method of Rational Univariate Representation (primitive element) [Canny,Rouillier] for isolating all real α .

Polynomial System Solving II “Hide” a variable in the coefficient field: f 0 , f 1 , . . . , f n ∈ ( K [ x 0 ])[ x 1 , x − 1 1 , . . . , x n , x − 1 n ] Hypothesis: x 0 -coordinates of roots distinct, | M ( x 0 ) | �≡ 0. � � � � M 12 ( x 0 ) M 12 ( x 0 ) M 11 M 11 � � � � det M ( x 0 ) = � = � , M ′ ( x 0 ) � � � � M 22 ( x 0 ) 0 M 21 � � 0 + · · · + A − 1 d A 1 x 0 + A − 1 | M ′ ( x 0 ) | = | A d x d 0 + · · · + A 1 x 0 + A 0 | = det A d det( x d d A 0 ) . • If det A d � = 0, define companion matrix C :   0 I 0 . ... . .   C =   0 0 I   − A − 1 − A − 1 − A − 1 d A 0 d A 1 · · · d A d − 1 The eigenvalues of C are the x 0 -coordinates of the solutions and its eigenvectors contain the values of the monomials indexing M ′ at the roots. • Rank balancing improves the conditioning (of A d ) by x �→ ( t 1 y + t 2 ) / ( t 3 y + t 4 ), t i ∈ R Z . • If A d remains ill-conditioned, solve the generalized eigenproblem     0 − I I ... ...      x +  .     − I I   · · · A d A 0 A 1 A d − 1

Matrix-based methods for system solving Let { z k } k ⊂ C n be the isolated zeros of f 1 , . . . , f n ∈ Theorem. Q [ x 1 , . . . , x n ]. There exists matrix M a expressing multiplication by a mod � f i � s.t. • the eigenvalues of M a are a ( z k ), and the eigenvectors of M t • a are, up to a scalar, 1 z k : p ( x ) �→ p ( z k ). Construct multiplication matrices by means of • resultant matrices, e.g. Sylvester, B´ ezout, sparse, or • normal forms, boundary bases (generalize Gr¨ obner bases). Stable with respect to input perturbations. Handles multiplicities and zero sets at infinity. Extends to over-constrained systems and 1-dimensional zero sets. Complexity: single exponential in n . Synaps/Mathemagix library: C++, fast univariate solvers (e.g. [E- Tsigaridas] ), connections (GMP, MPFR, LAPACK, SparseLU etc).

Multiplication maps Consider ideal I := � p 1 , . . . , p m � ⊂ K [ x 1 , . . . , x n ] = K [ x ], and quotient ring A K := K [ x ] /I . Polynomial multiplication in A K , ie. mod I , by some a ∈ K [ x ], is a linear map: M a : K [ x ] /I → K [ x ] /I : b �→ ab mod I, where ordered field K ⊂ some real closed field. Can compute M a via Resultants, Gr¨ obner bases, normal-form meth- ods.

Software

Discrete geometry • (Stable) Mixed Cells Input: n polynomial supports in Z n (well-constrained) Output: Monomial basis of quotient, generic number of roots in ( K ∗ ) n , K n , starting system of sparse homotopy in ( K ∗ ) n , K n . Code: Ansi-C. Package: MMX (SYNAPS) and stand-alone.

Symbolic algebra Input: n + 1 polynomial supports in Z n (over-constrained) Output: Square toric resultant matrix, optimal size in specified polynomial • Incremental algorithm Features: Exact Sylvester-type matrix whenever possible Code: Ansi-C. Package: MMX (SYNAPS) or stand-alone. Future work: Fast rank tests (quasi-Toeplitz matrices), MMX (SYNAPS) sparse representations (superLU, Hewlett), monomial set representation (+ arithmetic) • Subdivision-based (greedy) algorithm Features: Exact rational expression, allows linear perturbation. Code: Maple. Package: Multires or stand-alone. Future work: Sparse / structured representations, fast point-in-cell location (in implicit subdivision)

Numerical solving • Polynomial system solving Input: Polynomial supports and coefficients, resultant matrix Output: Superset of common roots Features: Numerical linear algebra: LAPACK, trade-off between speed and accuracy, factors out constant submatrix: Schur factorization, rank balancing of matrix polynomial, regular or generalized eigenproblem. Code: Ansi-C, some in Maple. Package: Stand-alone. Future work: • MMX (SYNAPS) capabilities: Popov, quasi-Toeplitz structure, arithmetic. • LAPACK capabilities: condition numbers, backward-error analysis.