Gröbner bases over Tate algebras Xavier Caruso 1 Tristan Vaccon 2 Thibaut Verron 3 1. Université de Bordeaux, CNRS, Inria, Bordeaux, France 2. Université de Limoges, CNRS, XLIM, Limoges, France 3. Johannes Kepler University, Institute for Algebra, Linz, Austria 16 July 2019, ISSAC 2019, Beijing, China 1
Precision and Gröbner bases ◮ Qestion: in R [ X ] , reduce f = X 2 modulo g = 0 . 01 X − 1 2
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 ◮ 2
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 ◮ 2
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 ◮ 2
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 ◮ 2
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 ◮ 2
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 ◮ 2
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 ]] 2
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: if a and b are small, a + b is small ? a + b = a + b a + b = a + b 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: ◮ In a CDVR, a series is convergent if a and b are small, a + b is small iff its general term tends to 0 ? a + b = a + b a + b = a + b � 0 n = 0 a n = a 0 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: ◮ In a CDVR, a series is convergent if a and b are small, a + b is small iff its general term tends to 0 ? a + b = a + b a + b = a + b � 1 n = 0 a n = a 0 + a 1 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: ◮ In a CDVR, a series is convergent if a and b are small, a + b is small iff its general term tends to 0 ? a + b = a + b a + b = a + b � 2 n = 0 a n = a 0 + a 1 + a 2 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: ◮ In a CDVR, a series is convergent if a and b are small, a + b is small iff its general term tends to 0 ? a + b = a + b a + b = a + b � 3 n = 0 a n = a 0 + a 1 + a 2 + a 3 3
A recap on Complete Discrete Valuation Rings ◮ DVR = principal local domain K ◦ with maximal ideal � π � , residue field F = K ◦ / � π � Z p p F p C [[ X ]] X C ◮ Elements can be writen a = � ∞ n = 0 a n π n , a n ∈ F ◮ Valuation of a = max n such that π n divides a ◮ Metric defined by “ a is small ⇐ ⇒ val ( a ) is large” val ( a ) = 3 ◮ Z p and C [[ X ]] are complete for this topology a = a 3 π 3 + a 4 π 4 + . . . 1 π ◮ No loss of precision possible: ◮ In a CDVR, a series is convergent if a and b are small, a + b is small iff its general term tends to 0 ? a + b = a + b a + b = a + b � ∞ n = 0 a n = a 0 + a 1 + a 2 + a 3 + · · · 3
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 4
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 5
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 + . . . 5
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