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A remark on the composition of polynomial functions over algebraically closed fields A remark on the composition of polynomial Erhard Aichinger and functions over algebraically closed fields Stefan Steinerberger Compositions Erhard Aichinger and Stefan Steinerberger An algebraic approach A multivariate Universität Linz and Universität Bonn generalization AAA81 Salzburg, February 2011

Compositions that are polynomial functions A remark on the composition of polynomial functions over algebraically closed fields Question Erhard Let K be a field, and let f , g : K → K . We assume Aichinger and Stefan Steinerberger f ◦ g is polynomial, Compositions g is polynomial. An algebraic Can we conclude that approach A multivariate generalization f is a polynomial on the range of g ?

Some observations A remark on the Obvious fact composition of polynomial functions over Let K be a finite field, and let f , g : K → K . If algebraically closed fields f ◦ g and g are polynomial, Erhard Aichinger and Stefan then Steinerberger f is polynomial. Compositions An algebraic approach A multivariate The real case generalization √ x , g ( x ) := x 3 . Then On the reals, let f ( x ) := 3 f ◦ g ( x ) = x for all x ∈ R , but f is not polynomial.

The complex numbers A remark on the composition of polynomial functions over algebraically Theorem closed fields Erhard Let f , g : C → C , g nonconstant polynomial, f ◦ g polynomial. Aichinger and Then f is polynomial. Stefan Steinerberger Sketch of the proof: Compositions By complex analysis arguments, f is holomorphic An algebraic approach [Rudin, 1966, Chapter 10, p.221, Exercise 20]. A multivariate generalization n If, for large | x | , | g | ∼ | x m | , | f ◦ g | ∼ | x n | , then | f | ∼ | x m | , and hence (Liouville) f is polynomial.

Some prerequisites A remark on Observation the composition of Let K be a field, f , g , h : K → K such that polynomial functions over algebraically h = f ◦ g . closed fields Erhard Aichinger and Then for all a , b ∈ K , we have Stefan Steinerberger g ( a ) = g ( b ) = ⇒ h ( a ) = h ( b ) . Compositions An algebraic approach Hilbert’s Nullstellensatz A multivariate generalization Let A be an algebraically closed field, and let f 1 , . . . , f m , g ∈ A [ x 1 , . . . , x n ] . TFAE: For all a ∈ A n : f A 1 ( a ) = · · · = f A m ( a ) = 0 = ⇒ g A ( a ) = 0 . ∃ r ∈ N 0 ∃ b 1 , . . . , b m ∈ A [ x 1 , . . . , x n ] : g r = b 1 · f 1 + · · · + b m · f m .

Fried and MacRae’s Theorem A remark on the composition of polynomial Theorem [Fried and MacRae, 1969] functions over Let K be a field, p , q , f , g ∈ K [ t ] , deg ( p ) > 0, deg ( q ) > 0. TFAE: algebraically closed fields p ( x ) − q ( y ) | f ( x ) − g ( y ) in K [ x , y ] . Erhard Aichinger and Stefan ∃ h ∈ K [ t ] : f = h ( p ( t )) and g = h ( q ( t )) . Steinerberger Compositions Proofs: An algebraic approach Original proof: field of algebraic functions over some curve. A multivariate generalization Elementary algebraic proofs by E.A. and F. Binder [Binder, 1996]. [Schicho, 1995]: J. Schicho provides a proof from category theory that suggests many generalizations.

From C to algebraically closed fields A remark on the composition of polynomial functions over algebraically closed fields Theorem Erhard Aichinger and Let A be an algebraically closed field, let f , g : A → A . If Stefan Steinerberger g is polynomial, f ◦ g is polynomial, g ′ � = 0, Compositions then An algebraic approach A multivariate f is polynomial. generalization

A multivariate generalization A remark on the composition of polynomial functions over algebraically Theorem closed fields Let A be an algebraically closed field, n ∈ N , p 1 , . . . , p n ∈ A [ t ] , Erhard Aichinger and and let f be a function from A n to A . We assume that Stefan Steinerberger g : A n → A , ( a 1 , . . . , a n ) �→ f ( p A 1 ( a 1 ) , . . . , p A n ( a n )) Compositions An algebraic is a polynomial function, and that for each i ∈ { 1 , . . . , n } , the approach derivative p ′ i � = 0. Then A multivariate generalization f is a polynomial function.

A proof by reduction to the unary case A remark on the composition of polynomial functions over algebraically closed fields Theorem [Prager and Schwaiger, 2009] Let K be a field with | K | > ℵ 0 , and let f : K n → K . If for all Erhard Aichinger and Stefan i ∈ { 1 , . . . , n } and all b 1 , . . . , b n ∈ K , Steinerberger x �→ f ( b 1 , . . . , b i − 1 , x , b i + 1 , . . . , b n ) Compositions An algebraic is a polynomial function, then approach A multivariate generalization f is a polynomial function.

A generalization of Fried and MacRae’s Theorem A remark on the composition of Theorem (cf. [Schicho, 1995]) polynomial functions over algebraically Let K be a field, n ∈ N , let p 1 , . . . , p n , q 1 , . . . , q n be nonconstant closed fields polynomials in K [ t ] , and let f , g ∈ K [ t 1 , . . . , t n ] . Then the following Erhard are equivalent: Aichinger and Stefan Steinerberger Compositions f ( x 1 , . . . , x n ) − g ( y 1 , . . . , y n ) ∈ � p i ( x i ) − q i ( y i ) | | | i ∈ { 1 , . . . , n }� K [ x , y ] . An algebraic approach A multivariate There is h ∈ K [ t 1 , . . . , t n ] such that generalization f ( x 1 , . . . , x n ) = h ( p 1 ( x 1 ) , . . . , p n ( x n )) g ( x 1 , . . . , x n ) = h ( q 1 ( x 1 ) , . . . , q n ( x n )) .

A generalization of “ p ( x ) − p ( y ) is squarefree”. A remark on the composition of polynomial functions over algebraically closed fields Lemma Erhard Aichinger and Let A be an algebraically closed field, and let Stefan Steinerberger p 1 , . . . , p n , q 1 , . . . , q n ∈ A [ t ] with p ′ i � = 0 and q ′ i � = 0 for all i . Then Compositions � p i ( x i ) − q i ( y i ) | | | i ∈ { 1 , . . . , n }� A [ x , y ] An algebraic approach is a radical ideal. A multivariate generalization Remark: “Algebraically closed” can be dropped.

A remark on Binder, F. (1996). the Fast computations in the lattice of polynomial rational composition of polynomial function fields. functions over algebraically In Lakshman, Y. N., editor, Proceedings of the 1996 closed fields Erhard international symposium on symbolic and algebraic Aichinger and computation, ISSAC ’96, Zuerich, Switzerland, July Stefan Steinerberger 24–26, 1996. New York, NY: ACM Press. 43-48. [ISBN Compositions 0-89791-796-0/pbk] . An algebraic approach Fried, M. D. and MacRae, R. E. (1969). A multivariate On curves with separated variables. generalization Math. Ann. , 180:220–226. Prager, W. and Schwaiger, J. (2009). Generalized polynomials in one and in several variables. Math. Pannon. , 20(2):189–208.

A remark on Rudin, W. (1966). the Real and complex analysis . composition of polynomial McGraw-Hill Book Co., New York. functions over algebraically closed fields Schicho, J. (1995). Erhard A note on a theorem of Fried and MacRae. Aichinger and Stefan Arch. Math. (Basel) , 65(3):239–243. Steinerberger Compositions An algebraic approach A multivariate generalization

A remark on the composition of polynomial functions over algebraically closed fields Erhard Aichinger and Stefan Steinerberger Compositions An algebraic approach A multivariate generalization