Geometry of Hilbert polynomials and Gröbner bases Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings John Perry Department of Mathematics, The University of Southern Mississippi Hilbert polynomials Caboara, Sturmfels’ algorithms Future February 2010 Algebra is merely geometry in words; geometry is merely algebra in pictures. — Sophie Germain Outline Geometry of Hilbert polynomials and Gröbner bases John Perry 1 Gröbner bases and monomial orderings Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 2 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 4 Future Hilbert polynomials Caboara, Sturmfels’ algorithms Future Outline Geometry of Hilbert polynomials and Gröbner bases John Perry 1 Gröbner bases and monomial orderings Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 2 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 4 Future Hilbert polynomials Caboara, Sturmfels’ algorithms Future Gröbner basis: definition from example Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings 4 3 Gröbner bases Geometry of orderings 2 Hilbert polynomials 1 Caboara, Sturmfels’ algorithms -0 -1 Future -2 -3 -4 -4 -3 -2 -1 -0 1 2 3 what can I say about • ? 4 Monomial orderings Geometry of Hilbert polynomials and Gröbner bases John Perry x + xy 2 . x2 + xy (lex) 2 Gröbner bases and monomial orderings & x2 + xy2 (tdeg) Gröbner bases Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms Future Remark • notation: lm (p) • uncountably many orderings • given an ideal, finitely many equivalence classes Monomial orderings Geometry of Hilbert polynomials and Gröbner bases John Perry x + xy 2 . x2 + xy (lex) 2 Gröbner bases and monomial orderings & x2 + xy2 (tdeg) Gröbner bases Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms Future Remark • notation: lm (p) • uncountably many orderings • given an ideal, finitely many equivalence classes Gröbner basis: precise definition G a Gröbner basis of ideal I? Geometry of Hilbert polynomials and Gröbner bases John Perry 4 Gröbner bases and monomial orderings Gröbner bases 3 Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms 1 2 hlm(I)i • I = 〈G〉. . . • 4 3 2 1 Future Gröbner basis: precise definition G a Gröbner basis of ideal I? Geometry of Hilbert polynomials and Gröbner bases John Perry 4 Gröbner bases and monomial orderings Gröbner bases 3 Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms 1 2 hlm(G)i 4 3 2 1 Future y3 6∈ lm (G) • I = 〈G〉 but • lm (I) 6= lm (G) Gröbner basis: precise definition G a Gröbner basis of ideal I? Geometry of Hilbert polynomials and Gröbner bases John Perry 4 Gröbner bases and monomial orderings Gröbner bases 3 Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms 1 2 hlm(G)i 4 3 2 1 Future y3 ∈ lm (G) • I = 〈G〉 and • lm (I) = lm (G) Gröbner basis: facts Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings • finite GB exists for any I • GB varies by equivalence class • • size, density, complexity unique “reduced GB” Hilbert polynomials Caboara, Sturmfels’ algorithms Future Computing Gröbner bases Geometry of Hilbert polynomials and Gröbner bases John Perry Buchberger’s algorithm (1965), others • while lm (G) 6= lm (I) : Gröbner bases and monomial orderings Gröbner bases 4 Geometry of orderings 3 Hilbert polynomials 2 Caboara, Sturmfels’ algorithms 4 3 2 1 1 Future add • to G • • “easy” iff lm (G) 6= lm (I) Minkowski sum of Newton polyhedra Geometry of Hilbert polynomials and Gröbner bases John Perry another view of orderings F = x + y − 4, xy − 1 2 2 Gröbner bases and monomial orderings Gröbner bases Geometry of orderings Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 2 Future 1 1 2 3 (vertex ← exponent vector of each u ∈ f1 · · · fm ) Big-time fact Normal cones of Minkowski sum of F l equivalence classes Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 4 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 2 Future 1 1 2 3 4 classification of orderings ←→ discrete geometry (Gritzmann and Sturmfels, 1993) Outline Geometry of Hilbert polynomials and Gröbner bases John Perry 1 Gröbner bases and monomial orderings Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 2 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 4 Future Hilbert polynomials Caboara, Sturmfels’ algorithms Future Hilbert function: definition from example Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings 2.5 2 Gröbner bases Geometry of orderings 1.5 1 Hilbert polynomials 0.5 -0 Caboara, Sturmfels’ algorithms -0.5 -1 -1.5 Future -2 -2.5 -2.5 -2 -1.5 -1 -0.5 -0 0.5 1 dimF (•)? 1.5 2 2.5 Hilbert function: precise definition R = F x1 , . . . , xn , homogeneous ideal Geometry of Hilbert polynomials and Gröbner bases John Perry Hilbert function: Gröbner bases and monomial orderings HFI : N → N Gröbner bases Geometry of orderings d → dimF (Rd /Id ) . Hilbert polynomials Caboara, Sturmfels’ algorithms Facts 3 4 3 2 1 1 2 • HFI = HF lm(I) 〈 〉 4 Future • ∃ HFI for all I dimF (Rd /Id ) = dimF R/ lm (I) d Hilbert polynomial Hilbert polynomial: Geometry of Hilbert polynomials and Gröbner bases John Perry HPI (d) = HFI (d) for “sufficiently large” d. Gröbner bases and monomial orderings Gröbner bases 4 Geometry of orderings 3 Hilbert polynomials 1 2 Caboara, Sturmfels’ algorithms Facts • ∃ HFI , HPI for all I • deg HPI = dimF (•) 4 3 2 1 Future HFI (d) ¡ GB (I) Geometry of Hilbert polynomials and Gröbner bases John Perry Know G = GB (I), not HFI ? Gröbner bases and monomial orderings • HF lm(G) = HFI 〈 〉 • “easy” to compute: Gröbner bases Geometry of orderings Hilbert polynomials HSI , power series expansion of HFI • HPI • Caboara, Sturmfels’ algorithms Future Know HFI , not GB (I)? • HFI (d) degree d mononomials not in lm (G) d ? ( ( • ( degree d( pairs ((( HFI (d) ¡ GB (I) Geometry of Hilbert polynomials and Gröbner bases John Perry Know G = GB (I), not HFI ? Gröbner bases and monomial orderings • HF lm(G) = HFI 〈 〉 • “easy” to compute: Gröbner bases Geometry of orderings Hilbert polynomials HSI , power series expansion of HFI • HPI • Caboara, Sturmfels’ algorithms Future Know HFI , not GB (I)? • HFI (d) degree d mononomials not in lm (G) d ? ( ( • ( degree d( pairs ((( Outline Geometry of Hilbert polynomials and Gröbner bases John Perry 1 Gröbner bases and monomial orderings Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 2 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 4 Future Hilbert polynomials Caboara, Sturmfels’ algorithms Future Problem statement Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases • GB varies by equivalence class • • size, density, complexity some orderings more efficient than others Is it advisable to change orderings while computing a Gröbner basis? Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms Future “Tentative” Hilbert functions Geometry of Hilbert polynomials and Gröbner bases John Perry ¬ ¶ J = lm Gh Gröbner bases and monomial orderings Gröbner bases Geometry of orderings Definition tentative Hilbert function of 〈G〉 is HFJ Hilbert polynomials Caboara, Sturmfels’ algorithms Future Example 1 Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings • Gh = x2 + y2 − 4h2 , xy − h2 • lm Gh = x2 , xy HSJ (t) = t2 − t − 1 −t2 + 2t − 1 HSI (d) = t2 + 2t + 1 −t + 1 Gröbner bases Geometry of orderings Hilbert polynomials HPJ (d) = d + 2 Caboara, Sturmfels’ algorithms Future HPI (d) = 4 Example 1 Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings • Gh = x2 + y2 − 4h2 , xy − h2 • lm Gh = x2 , xy HSJ (t) = t2 − t − 1 −t2 + 2t − 1 HSI (d) = t2 + 2t + 1 −t + 1 Gröbner bases Geometry of orderings Hilbert polynomials HPJ (d) = d + 2 Caboara, Sturmfels’ algorithms Future HPI (d) = 4 Example 2 Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases G = x3 y2 + 5776x2 y3 + · · ·, x4 y + · · · tdeg, x > y tdeg, x < y HPJ (d) = 4d − 1 HPJ (d) = 3d + 4 Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms Future Which ordering “better”? Geometry of Hilbert polynomials and Gröbner bases John Perry If G not GB, 5 5 4 4 3 3 2 2 1 1 Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 1 2 3 4 R/ lm (G) 5 Hilbert polynomials Caboara, Sturmfels’ algorithms 1 2 3 4 R/ lm (I) HFI (d) = dimF (R/I)d ∴ minimize HF〈lm(G)〉 (d) in long run 5 Future Example Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings HPI 3d + 4 4d − 1 0 4 -1 1 7 3 2 10 7 d 3 4 13 16 11 15 5 19 19 6 22 23 #GB 5 15 Hilbert polynomials Caboara, Sturmfels’ algorithms Future Gritzmann-Sturmfels algorithm Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings “Dynamic” Buchberger algorithm: Gröbner bases Geometry of orderings • after adding new polynomial: • • Hilbert polynomials compute HF for each normal cone select ordering w/“best” HF Caboara, Sturmfels’ algorithms Future Gritzmann-Sturmfels, 1993 Caboara algorithm Geometry of Hilbert polynomials and Gröbner bases John Perry Gritzmann-Sturmfels algorithm: Gröbner bases and monomial orderings • compute HF for each normal subcone Gröbner bases Geometry of orderings Hilbert polynomials Observations • trade-off in complexity • • Caboara, Sturmfels’ algorithms most timings improve some worsen Future • only implementation of Gritzmann-Sturmfels algorithm? Caboara, 1993 Caboara algorithm Geometry of Hilbert polynomials and Gröbner bases John Perry Gritzmann-Sturmfels algorithm: Gröbner bases and monomial orderings • compute HF for each normal subcone Gröbner bases Geometry of orderings Hilbert polynomials Observations • trade-off in complexity • • Caboara, Sturmfels’ algorithms most timings improve some worsen Future • only implementation of Gritzmann-Sturmfels algorithm? Caboara, 1993 Outline Geometry of Hilbert polynomials and Gröbner bases John Perry 1 Gröbner bases and monomial orderings Gröbner bases and monomial orderings Gröbner bases Geometry of orderings 2 Hilbert polynomials 3 Caboara, Sturmfels’ algorithms 4 Future Hilbert polynomials Caboara, Sturmfels’ algorithms Future Open question Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Gröbner bases Geometry of orderings • small penalty to compute HF • diminishing benefit from “better” order • when quit computing HF? • guess: when HFJ (d) stable up to largest degree pair? Hilbert polynomials Caboara, Sturmfels’ algorithms Future Finis Geometry of Hilbert polynomials and Gröbner bases John Perry Gröbner bases and monomial orderings Thank you! • LATEX • • Beamer • • audience Gröbner bases Geometry of orderings Hilbert polynomials Caboara, Sturmfels’ algorithms Future
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