Tropical bases by regular projections Kerstin Hept Eindhoven, - PowerPoint PPT Presentation

✁ ✄ Tropical bases by regular projections Kerstin Hept Eindhoven, 10/31/07 joint work with Thorsten Theobald www.arXiv.org/abs/0708.1727 •••••••••••••••••••••••••••••••••••

Topics ✁ ✄ First session: Second session: • Tropical hypersurface • The main theorem • Newton polytope and its subdi- • Construction of appropriate vision projections • Properties of the tropical hyper- • Preimage of a projection surface • Example • Real valuations, Puiseux series • Tropicalization of a polynomial • Tropical variety and prevariety and their properties • Tropical bases • Linear spaces •••••••••••••••••••••••••••••••••••

Tropical Semiring ✁ ✄ The main algebraic structure in tropical geometry is the tropical semiring: Definition. The tropical semiring is the triple ( R ∪{∞} , ⊕ , ⊙ ) with the following tropical addition and multiplication a ⊕ b := min( a, b ) a ⊙ b := a + b We observe: • The compositions are commutative, associative und distributive. • ∞ is the additive and 0 is the multiplicative neutral element • There is NO TROPICAL SUBTRACTION. •••••••••••••••••••••••••••••••••••

Tropical polynomial ring ✁ ✄ Tropical polynomial ring in the unknowns x 1 , . . . , x n : Monomials: x ⊙ i 1 ⊙ x ⊙ i 2 ⊙ . . . ⊙ x ⊙ i n n 1 2 i 1 , . . . , i n ∈ Z Polynomials: Finite linear combinations of monomials a ⊙ x i 1 1 ⊙ x i 2 2 ⊙ . . . ⊙ x i n p ( x 1 , . . . , x n ) = n ⊕ b ⊙ x j 1 1 ⊙ x j 2 2 ⊙ . . . ⊙ x j n n ⊕ . . . = min { a + i 1 x 1 + . . . + i n x n , b + j 1 x 1 + . . . + j n x n , . . . } •••••••••••••••••••••••••••••••••••

Tropical hypersurface ✁ ✄ Definition. Let p ( x 1 , . . . , x n ) be a tropical polynomial. Then { x ∈ R n | T ( p ) := the minimum is attained at least twice in x } is the tropical hypersurface defined by p . Example. Tropical curves in the plane q := 1 ⊕ 0 ⊙ x 1 ⊕ 0 ⊙ x 2 ⊕ 1 ⊙ x 2 1 ⊕ 0 ⊙ x 1 ⊙ x 2 ⊕ 1 ⊙ y 2 p := 0 ⊙ x 1 ⊕ 1 ⊙ x 2 ⊕ 2 � � � � � � � � � � � � � Figure: a tropical line Figure: a tropical quadratic curve •••••••••••••••••••••••••••••••••••

Newton polytope ✁ ✄ Definition. Let f be a tropical polynomial in the unknowns x 1 , . . . , x n with terms x i 1 1 ⊙ . . . ⊙ x i n n . Then the Newton polytope of f is the convex hull New( f ) := conv { ( i 1 , . . . , i n ) ∈ R n : x i 1 1 · · · x i n n a term in p } Example. f = 2 ⊙ x 2 ⊕ x 2 ⊙ y 3 ⊕ 3 ⊙ x ⊙ y 2 ⊕ 1 ✻ 3 � � � 2 ✁ ✁ ✁ 1 ✁ ✁ ✁ ✲ 1 2 Figure: the Newton polytope of f •••••••••••••••••••••••••••••••••••

Subdivision ✁ ✄ Definition. For a tropical polyno- mial p the extended Newton polytope is: conv { ( i 1 , . . . , i n , c i ) : c i ⊙ x i 1 1 ⊙ . . . ⊙ x i n n a monomial in p } The projection of the lower bound onto the first n coordinates gives a subdivision of the Newton polytope. The tropical hypersurface is dual to that subdivision. •••••••••••••••••••••••••••••••••••

Examples ✁ ✄ Example. A subdivided Newton polytope and its dual. p := 3 ⊙ x 2 1 ⊕ 2 ⊙ x 1 ⊙ x 2 ⊕ 3 ⊙ x 2 2 ⊕ 0 2 3 ❅ ❅ � ❅ ❅ 2 ❅ � � � ❅ � ❅ 2 � � ❅ 3 0 � � Example. One can recognize the type of the tropical hypersurface only by analysing the subdivided Newton polytope. ❅ � ❅ � ❅ ❅ � ❅ ❅ � � ❅ � ❅ � � � •••••••••••••••••••••••••••••••••••

Balancing condition ✁ ✄ Let e be an edge of a tropical curve T ( f ) in R 2 . Then one can define its multiplicity by the lattice length of the corresponding edge in the subdivided Newton polytope. Let p be any vertex of T ( f ), v 1 , . . . , v r the primitive lattice vectors in the direction of the edges emanating from p and m 1 , . . . , m r the corresponding multiplicities. Then the following balancing condition holds: m 1 · v 1 + m 2 · v 2 + . . . + m r · v r = 0 (This holds because of the duality) 2 � ❅ ■ � ✒ � ❅ � 2 · (0 , − 1) + (1 , 1) + ( − 1 , 1) = 0 2 2 ❄ � � 2 •••••••••••••••••••••••••••••••••••

Real valuations ✁ ✄ For a field K , a real valuation is a map ord : K → ¯ R = R ∪ {∞} with • K \ { 0 } → R and • 0 �→ ∞ • ord( ab ) = ord( a ) + ord( b ) and • ord( a + b ) ≥ min { ord( a ) , ord( b ) } . Example. K = Q with the p -adic valuation: Every q ∈ Q with q = p s m n , p � | m, p � | n, s ∈ Z has the valuation ord( q ) = v p ( q ) := s. K n via We can extend the valuation map to an algebraic closure ¯ K and then to ¯ K n → ¯ ord : ¯ R n , ( a 1 , . . . , a n ) �→ (ord( a 1 ) , . . . , ord( a n )) . •••••••••••••••••••••••••••••••••••

Puiseux series ✁ ✄ Another interesting field is C {{ t }} , the field of puiseux series: The elements are formal power series • with coefficients in C . • with rational exponents, bounded below with a common denominator C {{ t }} is an algebraic closed field. A valuation is given by the order map ord : ord : C {{ t }} → R c α x α �→ min { α | c α � = 0 } � α ∈A •••••••••••••••••••••••••••••••••••

Tropicalization ✁ ✄ α c α x α be a polynomial in K [ x 1 , . . . , x n ] where K is a field with a Let f = � valuation ord. Definition. The tropicalization of f is defined as � ord( c α ) ⊙ x α trop( f ) := α � ord( c α ) ⊙ x α 1 1 ⊙ . . . ⊙ x α n = n α = min α { ord( c α ) + α 1 x 1 + · · · + α n x n } and the tropical hypersurface of f is T ( f ) := T ( trop ( f )) { w ∈ R n : the minimum in trop( f ) = is attained at least twice in w } •••••••••••••••••••••••••••••••••••

✁ ✄ Example. Let f = 2 x + 4 y − x 2 + 3 y 2 a polynomial in Q [ x, y ] with the 2 -adic valuation. Then 1 ⊙ x ⊕ 2 ⊙ y ⊕ 0 ⊙ x 2 ⊕ 0 ⊙ y 2 trop ( f ) = = min { 1 + x, 2 + y, 2 x, 2 y } ✻ 3 � � ✟� 2 ✟✟✟✟✟ 1 � � ✲ � � 1 2 3 � � Figure: The tropical hypersurface T ( f ) •••••••••••••••••••••••••••••••••••

Tropical variety ✁ ✄ We generalize that to ideals: Definition. For an ideal I ✁ K [ x 1 , . . . , x n ] , the tropical variety of I is defined by � T ( I ) = T ( f ) f ∈ I There is another description of the tropical variety of an ideal: Proposition (Kapranov). If the valuation is nontrivial, then the tropical variety is the topological closure T ( I ) = ord V ( I ) K ∗ ) n is the variety of I . where V ( I ) ⊂ ( ¯ •••••••••••••••••••••••••••••••••••

Example ✁ ✄ Let I be generated by f 1 := 2 + y − 4 x 2 y + x 2 y 2 + 2 xy 2 f 2 := xyz − 2 z + 4 xyz 2 − 2 + z 2 Then we get the tropical variety in R 3 : Figure: Tropical variety T ( I ) •••••••••••••••••••••••••••••••••••

Properties ✁ ✄ A tropical variety has several properties: Proposition (Bieri-Groves 1984). Let I be a prime ideal. Then T ( I ) is a pure m -dimensional complex where m = dim( I ) is the Krull dimension of the ideal.. Proposition (Bieri-Groves 1984). T ( I ) is totally concave, which means that each convex hull of a local cone of a point x is an affine subspace. ❅ � ❅ � ❅ � � ❅ ■ � ✒ ❅ � ❅ � � � ❅ � ❅ � LC x ( T ( I )) x ❄ � � � Speyer showed in 2005 that the balancing condition, which is stronger than the concavity condition, holds for tropical varieties in general. •••••••••••••••••••••••••••••••••••

Tropical prevariety ✁ ✄ Definition. Let f 1 , . . . , f s ∈ K [ x 1 , . . . , x n ] be polynomials. Then the tropical pre- variety defined by f 1 , . . . , f s is the intersection of the tropical hypersurfaces s � T ( f 1 , . . . , f s ) := T ( f i ) i =1 This is in general not a tropical variety: Example. f 1 := t + ( t + 1) x 2 + (2 t 3 − t 4 ) y 2 + txy 1 3 3 2 + t 2 ) x + t 2 y f 2 := t + ( t ( 1 2 , − 1 2 ) � � ❅ � � ❅ ❅ � � � (1 , − 1) � � � � � � � � •••••••••••••••••••••••••••••••••••