Working with valuations Thibaut Verron Johannes Kepler University, - PowerPoint PPT Presentation

Working with valuations Thibaut Verron Johannes Kepler University, Institute for Algebra, Linz, Austria Séminaire Calcul Formel, Limoges 27 février 2020 1

Part 1: Signature Gröbner bases over Tate algebras joint work with Xavier Caruso 1 and Tristan Vaccon 2 1. Institut de Mathématiques de Bordeaux, Bordeaux, France 2. XLIM, Université de Limoges, Limoges, France Séminaire Calcul Formel, Limoges 27 février 2020 2

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 01 X − 1 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 01 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 100 Xg 0 . 01 X 3 100 X + 0 . 01 X 3 g − 10 000 g 0 . 0001 X 4 10 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 0001 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 10 000 Xg 0 . 0001 X 3 10 000 X + 0 . 0001 X 3 g − 100 000 000 g 0 . 000 000 01 X 4 100 000 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 000 001 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 1 000 000 Xg 0 . 000 001 X 3 1 000 000 X + 0 . 000 001 X 3 g − 1 000 000 000 000 g 0 . 000 000 000 001 X 4 1 000 000 000 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 01 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 100 Xg 0 . 01 X 3 100 X + 0 . 01 X 3 g − 10 000 g 0 . 0001 X 4 10 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 0001 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 10 000 Xg 0 . 0001 X 3 10 000 X + 0 . 0001 X 3 g − 100 000 000 g 0 . 000 000 01 X 4 100 000 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 000 001 X − 1 LT ( g ) ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 1 000 000 Xg 0 . 000 001 X 3 1 000 000 X + 0 . 000 001 X 3 g − 1 000 000 000 000 g 0 . 000 000 000 001 X 4 1 000 000 000 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ 3

Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 000 001 X − 1 ◮ The usual way: ◮ Another way? f = X 2 f = X 2 + X 2 g − 1 000 000 Xg 0 . 000 001 X 3 1 000 000 X + 0 . 000 001 X 3 g − 1 000 000 000 000 g 0 . 000 000 000 001 X 4 1 000 000 000 000 ◮ It terminates, but... · · · · · · ◮ g ≃ 1, but f mod g �≃ 0 It does not terminate, but... ◮ The sequence of reductions tends to 0 ◮ ◮ This work: make sense of this process for convergent power series in Z p [[ X ]] 3

Valued fields and rings: basic definitions Valuation: function val : k → Z ∪ {∞} with: ◮ val ( a ) = ∞ ⇐ ⇒ a = 0 0 ◮ val ( ab ) = val ( a ) + val ( b ) a · b = ab ? ? ? ◮ val ( a + b ) ≥ min( val ( a ) , val ( b )) a + b = a + b a + b = a + b val ( a ) = 3 a = a 3 π 3 + a 4 π 4 + . . . b = b − 3 π − 3 + b − 2 π − 2 + . . . π Examples: 1 val ( b ) = − 3 4

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � 0 n = 0 a n = a 0 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � 1 n = 0 a n = a 0 + a 1 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � 2 n = 0 a n = a 0 + a 1 + a 2 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � 3 n = 0 a n = a 0 + a 1 + a 2 + a 3 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � ∞ n = 0 a n = a 0 + a 1 + a 2 + a 3 + · · · 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � ∞ n = 0 a n = a 0 + a 1 + a 2 + a 3 + · · · 5

Examples of valued fields and rings Frac Ring K ◦ Field K Uniformizer π Residue field K ◦ /π Complete val ≥ 0 Z ( p ) Q p prime F p × Now Z p Q p p prime F p � C [ x ] ( x − α ) C ( x ) x − α C × C [[ x − α ]] C (( x − α )) x − α C � In part 2 ◮ Metric and topology defined by “ a is small” ⇐ ⇒ “val ( a ) is large” ◮ In a complete valuation ring, a series is convergent iff its general term goes to 0: � ∞ n = 0 a n = a 0 + a 1 + a 2 + a 3 + · · · 5

Tate Series X = X 1 , . . . , X n Definition ◮ K { X } ◦ = ring of series in X with coefficients in K ◦ converging for all x ∈ K ◦ = ring of power series whose general coefficients tend to 0 Motivation ◮ Introduced by Tate in 1971 for rigid geometry ( p -adic equivalent of the bridge between algebraic and analytic geometry over C ) Examples ◮ Polynomials (finite sums are convergent) ∞ π i + j X i Y j = 1 + π X + π Y + π 2 X 2 + π 2 XY + π 2 Y 2 + · · · � ◮ i , j = 0 ∞ X i = 1 + 1 X + 1 X 2 + 1 X 3 + · · · � ◮ Not a Tate series: i = 0 6

Term ordering for Tate algebras X i = X i 1 1 · · · X i n n ◮ Starting from a usual monomial ordering 1 < m X i 1 < m X i 2 < m . . . ◮ We define a term ordering puting more weight on large coefficients Usual term ordering: Term ordering for Tate series: π · 1 < m 1 X i 1 < m π X i 2 < m π 2 X i 3 < m · · · · · · < π 2 X i 3 < π · 1 < π X i 2 < 1 X i 1 < · · · < m 7

Term ordering for Tate algebras X i = X i 1 1 · · · X i n n ◮ Starting from a usual monomial ordering 1 < m X i 1 < m X i 2 < m . . . ◮ We define a term ordering puting more weight on large coefficients Usual term ordering: Term ordering for Tate series: π · 1 < m 1 X i 1 < m π X i 2 < m π 2 X i 3 < m · · · · · · < π 2 X i 3 < π · 1 < π X i 2 < 1 X i 1 < · · · < m ◮ It has infinite descending chains, but they converge to zero LT ( f ) ◮ Tate series always have a leading term f = a 2 XY + a 1 X + a 0 · 1 + a 3 X 2 Y 2 + . . . 7