The schedule and extra material of the
Problem Solving Seminar in Commutative Algebra
(and other mathematics)
This seminar has a format very different from the usual seminar activities. Instead of listening to talks on sporadic topics, we in the first place try to brainstorm and (hopefully) solve concrete problems suggested by the seminar participants. At the moment we are actively discussing different aspects and questions around the Waring problem for polynomials.
The important aspects of suggested problems should be an easy elegant formulation snd relevance to mathematics in general. We are trying to avoid complicated technical language which usually shadows the content of good problems. Everybody who wants to join us is very welcome.
In Fall 2015 we meet on Tuesdays at 10.00 in room 34, bldn 5, Kräfriket.
Notice that below you will find a lot of unpublished material for internal use only. If you want to spread this, please,
send a message to shapiro@math.su.se or ralff@math.su.se. This site will be regularly updated. We hope you will find
something interesting below.
- Problems with polynomials - the good, the bad, and the ugly, B.Shapiro, Arnold Mathematical Journal: Volume 1, Issue 1 (2015), Page 91--99
- Secants of powers, by R.Fröberg, A.Oneto, G.Ottaviani, B.Shapiro
- On Vandermonde varieties, by R.Fröberg, B.Shapiro
- List of unsolved problems and conjectures related to the Waring problem for polynomials, under permanent construction
Seminar Schedule, Fall 2016
- September 13, 13.00 rm. 16, V.Welker "Betti numbers of monomials ideals with DGA
structure on their minimal free resolution."
- September 27, 13.00 rm. 16, M.Boij "Lefschetz properties and powers of ideals."
- October 11, 13.00 rm. 16, T.Sadykov "Hypergeometric polynomials are optimal."
- October 25, 13.00 rm. 16, A.Kirillov "Om quadratic algebras."
- November 1, 13.00 rm. 16, Nasrin Altafi Razlighi "Low codimensional lattice ideals."
- December 16, 13.00 rm. 16, M.Shapiro, Michigan State University
"Cluster algebras compatible with Belavin-Drinfeld Poisson-Lie brackets"
Abstract: We discuss the progress in the conjecture that coordinate ring of SL_n has cluster structure compatible with any Poisson-Lie bracket from the famous Belavin-Drinfeld classification.
Seminar Schedule, Spring 2016
- January 12, 13.00. rm.34, E. Carlini, "Hadamard product of linear spaces"
- January 19, 13.00. rm.34, G. Nenashev, "A generalised super Fermat problem for binary forms"
- January 26, 13.00. rm.34, A. Oneto, "New results about the Waring rank of monomials"
- February 02, 13.00. rm.34, M.Boij, "???"
- February 09, 13.00. rm.34, L.Nickalsson, "A note on the Lefschetz properties in positive characteristic"
- February 16, 13.00. rm.34, R.Bögvad, "???"
- February 23, 13.00. rm.34, D.Novikov, "???"
- March 22, 13.00. rm.34, R. Fröberg, "POWERS VS SYMBOLIC POWERS FOR IDEAL OF POINTS IN P^n"
Seminar Schedule, Fall 2015
- September 15, 10.00. rm.34, S. Lundgvist, "On the zero set of a multivariate polynomial with coefficients from {0,1} over a field of prime order"
- September 22, 10.15. rm.34, R.Fröberg "Results about Hilbert functions of graded ideals
in the polynomial ring and in the exterior algebra."
- September 29, 10.15. rm.34, G.Nenashev "On Postnikov-Shapiro algebras and their generalisations."
- October 6, 10.15. rm.34, A.Oneto "On Waring-like decomposition of polynomials"
- October 13, 10.15. rm.34, R.Fröberg "Problem session."
- October 20, 10.15. rm.34, B.Shapiro "Problem session."
- November 10, 10.15. rm.34, K.Ranestad "The Waring problem for homogeneous and bihomogeneous forms in few variables, old and new results."
- November 17, 10.15. rm.34, E.Postinghel "Fat points in projective spaces: a geometric approach."
- November 24, 10.00. rm.34, G.Nenashev "A note on Fröberg's conjecture."
- December 2, 10.15. rm.34, D.Khavinson "Fingerprinting the lemniscate."
Seminar Schedule, Spring 2015
- January 22, 10.15. rm.16, R.Fröberg "Lefschetz properties of graded algebras"
- January 29, 10.15. rm.16, G.Ottaviani "Unique decomposition of tensors and the Waring problem"
- Ferbruary 5, 10.15. rm.16, B.Shapiro "On some new canonical forms for polynomials, after B.Reznick"
- February 12, 10.15. rm.16, L.Nicklasson "Some remarks about Fröberg's conjecture"
- March 5, 10.15. rm.16, B.Shapiro "On the zeros of positive semidefinite forms, after Reznick-Choi-Lam"
- March 12, 10.15. rm.16, B.Shapiro "On the zeros of positive semidefinite forms, after Reznick-Choi-Lam", continuation.
- March 19, 10.15. rm.16, R.Fröberg "Gorenstein rings, perp ideals, and inverse systems"
- March 26, 10.15. rm.16, M.Boij "On regular sequences of homogeneous ideals"
- April 1, 10.15. rm.16, L.Nicklasson "Another generalization of Mason's ABC-theorem following Michiel de Bondt"
- April 17, 10.15. rm.16, L.Nicklasson "Another generalization of Mason's ABC-theorem following Michiel de Bondt, cont."
- April 24, 10.15. rm.16, M.Boij "On perpideals of generic forms"
- May 8, 10.15. rm.16, R.Fröberg "Revision of a number of posed problems"
- May 20, 10.15. rm.16, A.Oneto "Waring locus of homogeneous polynomials"
Seminar Schedule, Fall 2014
- November 13, 10.15. rm.16, R.Fröberg "Numerical semigroups"
- November 20, 10.15. rm.16, R.Fröberg "A crash-course in Macaulay 2"
- November 27, 10.15. rm.16, A.Oneto "On the Waring rank of monomials"
- December 4, 10.15. rm.16, G.Nenashev "New commutative algebras associated to graphs"
- December 11, 10.15. rm.16, A.Oneto "On a class of power ideals"
- December 18, 10.15. rm.16, R.Fröberg "Curves and semigroups"
Some publications related to the Waring problem,
Fermat equations over functional fields, and Fröberg's conjecture,
in the inverse chronological order
- C.Scheiderer
"Sum of squares length of real forms", Preprint, March 2016.
- G. Blekherman, R. Sinn
"REAL RANK WITH RESPECT TO VARIETIES", Preprint, November 2015.
- A. de Paris,
"THE ASYMPTOTIC LEADING TERM FOR MAXIMUM RANK OF TERNARY FORMS OF GIVEN DEGREE", Preprint, October 2015.
- C.Goel, S.Kuhlmann, B. Reznick,
"The analogue of Hilbert's 1888 theorem for even symmetric forms", Preprint, September 2015.
- A. Kunert, C.Scheiderer
"EXTREME POSITIVE TERNARY SEXTICS", Preprint, August 2015.
- S.Friedland,
"Remarks of the symmetric rank of symmetric tensors", Preprint, May, 2015.
- A.Abdelmalek, Ch.Ikenmeyer, G.Royle,
"Lower bounds for sums of powers of lower degree univariates", Preprint, April 2015.
- N.Kayal, P.Koiran, T.Pecatte, Ch.Saha
"16,051 FORMULAS FOR OTTAVIANI?S INVARIANT OF CUBIC THREEFOLDS", Preprint, May 2015.
- E.Ballico, O.Dumitrescu, E.Postinghel,
"On Segre's bound for fat points in P^n", Preprint, April. 2015.
- L.Chiantini, G.Ottaviani, N. Vannieuwenholen,
"On generic identifiability of symmetric tensors of subgeneric rank", Preprint, April 2015.
- J.Buczynski, Z. Teitler,
"Some examples of forms of high rank", Preprint, April. 2015.
- M.V. Catalisano, A.V. Geramita, A. Gimighliano, B. Harbourna, J. Migliore, U. Nagel, and Y.S. Shin,
"Secant varieties of the varieties of reducible hyper surfaces in P^n", Preprint, Jan. 2015.
- T.Pecatte,
"Lower bounds for univariate polynomials: a Wronskian approach", Preprint, Nov. 2014.
- M.C. Brambilla, O.Dumitrescu, and E.Postinghel, "On linear systems of P^3 with nine base points", Preprint, Oct 2014.
- G.Blekherman and Z.Teitler, "On maximum, typical, and generic ranks", Preprint, Nov. 2014.
- B.Reznick and J.Rouse, "On sums of two cubes", Preprint, Nov. 2014.
- De Paris, " A proof that the maximal rank for plane quartics is 7", (2014) .
- Z.Teitler, "Geometric lower bounds for generalized ranks", (2014) .
- W.K. Hayman, "Waring's theorem and the super Fermat problem for numbers and functions", Comp. Var. and Ell. Eq. 59(1) (2014), 85-90.
- S. Iliman, T. de Wolff "Amoebas, nonnegative polynomials and sums of squares supported on circuits", preprint March 2014.
- G.Blekherman, G.Smith, M.Velasco "Sums of squares and varieties of minima degree", (2013) .
- A.Bernardi, "Normal bundle of rational curves and Waring decomposition", (2013) .
- M-C.Brambilla, O.Dumitrecu, E.Postinghel, "On a notion of speciality of linear systems in P^n", (2013) .
- S.Cooper, B.Harbourne, "Regina lectures on fat points", (2013) .
- E.Ballico, "On the typical rank of real polynomials (or symmetric tensors) with a fixed border rank", (2013) .
- Z.Teitler, "Maximum Waring ranks of monomials", (2013) .
- B.Reznick, "Some new canonical forms for polynomials", Pacific J.Math. 266(1) (2013) 185-220.
- E.Carlini, A.Oneto, "Monomials as sums of kth-powers of forms",
- K.Schmüdgen, "Around Hilbert's 17th Problem", Documenta Mathematica, Extra Volume ISMP (2012) 433--438
- A.Parab, "The ABC-conjecture for polynomials", Preprint, 2012.
- M.Bayat, H.Teimoori and M.Hassani, "An extension of ABC-theorem", Preprint, 2012.
- E.Ballico, M.C.Brambilla, F.Caruso, M.Sala, "Postulation of general quintuple fat points schemes of P^3", Preprint, 2011.
-
K.Ranestad, J-O.Schreyer, "Varieties of sums of powers", Preprint, Oct. 2011.
-
E.Carlini, M.V. Catalisano, and A.V. Geramita, "The solution to Waring's problem for monomials", Preprint, Oct. 2011.
- P. Pollack and C.Pomerance, "Fermat's last theorem for polynomials, lecture notes
- B.Reznick, "Linearly dependent powers of quadratic forms Preliminary report: 1999-2011", slides.
- B.Reznick, "On length of binary forms". 2012
-
C.Gasbarri, "Lectures on the ABC conjecture over function fields". 2010
-
C.Hillar, "Rational sums of squares with applications". 2010
- A. Causa, R. Re, "On the maximum rank of a real binary form". June 2010
- K.Pardue, "Generic sequences of polynomials". J.Algebra 324 (2010) 579--590.
- T. Chinburg, "Infinite easier Waring constants for commutative rings". June 2010
- D-I. Kim, "Fermat-type equations for Mbius transformations", Korean J. Math. 18(1) (2010), 29-35.
- S. Cooper, B. Harbourne, and Z.Teitler, "Combinatorial bounds on Hilbert functions of fat points in projective space". December 2010,
- A.Gimigliano, B.Harbourne, M.Ida, "Betti numbers for fat point ideals in the plane: geometric approach". TAMS, Volume 361, Number 2, February 2009, Pages 1103--1127
- F.Ardilla, A.Postnikov, "Combinatorics and geometry of power ideals". TAMS 2010
- A.Geramita, B.Harbourne, J.Migliore, "Classifying Hilbert functions of fat point subschemes in P^2". Collect. Math. 60, 2 (2009), 159--192
- M. de Bondt, "Another generalization of Mason's ABC-theorem". 2009
- P.Comon, G.Ottaviani, "On the typical rank of real binary forms". 2009
- E. Croot, D. Hart, "On sums and products in C[x]", Preprint, Apr. 2009
- J.Brachat, P.Comon, B. Mourrain, and E. P. Tsigaridas, "Symmetric tensor decomposition", Preprint, Jan. 2009
- S. Hu, "Sum of degrees of irreducible factors for polynomials in several variables", J.Math. Anal. Appl, 346 (2008) 340--343.
- M.C.Brambilla, G.Ottaviani, "On the Alexander-Hirschowitz theorem", J.Pure Appl. Algebra, 212 (2008) 1229--1251.
- C. Ciliberto, "Geometric Aspects of Polynomial Interpolation in More Variables and of Waring's Problem", preprint 2008
- L. Gallardo, L.Vaserstein, "The strict Waring problem for polynomial rings", article J. Number Theory, 128 (2008) 2963-2972
- M.Mella, "Base loci of linear systems and the Waring problem", preprint 2007
- K.Chandler, "The geometric interpretation of Fröberg-Iarrobino conjectures on infinitesimal neighborhoods
of points in projective space", J. Algebra 286 (2005) 421--455
- Yu-Ru Liu, T.D. Wooley, "Waring's problem in function fields", preprint, 2005.
- A.Geramita, "Secant Varieties and Inverse Systems", preprint, 2005.
- D.I.Kim, "Waring's problem for linear polynomials and Laurent polynomials", Rocky Mountain J. Math., 35 (5) (2005) 1533--1553.
- I.Dolgachev, "Dual Homogeneous Forms and Varieties of Power Sums", MILAN J MATH 72 (2004), 163-187
- A.Geramita, H.Schenck, "Fat Points, Inverse Systems, and Piecewise Polynomial Functions ", preprint, 2004.
- G.Gundersen, W.Hayman, "The strength of Cartan's version of Nevanlinna theory", Bull. LMS. 36 (2004) 433-454.
- E.Carlini, "Varieties of Simultaneous Sums of Powers for binary forms", preprint, 2002.
- E.Carlini, J.Chipalkatti, " On Waring's problem for several algebraic forms", preprint, 2001.
- R.C.Vaughan, T.D. Wooley, "Waring's Problem: A Survey", preprint, 2000.
- A.Iarrobino, "Inverse system of a symbolic power, III",
- A.Iarrobino, "Inverse system of a symbolic power, II", J. Algebra. 174 (1995) 1091-1110.
- J.Emsalem, A.Iarrobino, "Inverse system of a symbolic power, I", J. Algebra. 174 (1995) 1080-1090.
- M.Aubry, "Série de Hilbert d'une algébre de polynom quotient", J. Algebra. 176 (1995) 392-416.
- J.Alexander, and A.Hirschowitz, "Polynomial interpolation in several variables", J. Algebraic Geom. 4 (1995) 201-222.
- R.Fröberg, and J.Hollman, "Hilbert series for ideals generated by generic forms", J. Symbolic Comp. 17 (1994) 149-157.
- H.N.Shapiro, and G.Sparer, "Extension of a theorem of Mason", (1992).
- E.Bombieri, and J.Mueller, "The generalized Fermat equation in function fields", J.Number Theory 39, 339-350 (1991).
- V.Vassiliev, "Functions that are interpolating any k points", Funct. Anal. Appl, 1989,
- R.Fröberg, "An inequality for Hilbert series of graded algebras", Math.Scand 56 (1985) 117--144.
- J.Kung, G-C. Rota, "The invariant theory of binary forms", Bull AMS, vol. 10(1) (1984) 27--85
- W.Hayman, "Waring's Problem fuer analytische Funktionen", Bayerische Akademie der Wissenschaft, 1984.
- A.Geramita, F.Orecchia, "On the Chen-Macaulay Type of s-lines in A^{n+1}", J. ALgebra 70, 116-140 (1981)
- P.Riebenboim, "13 Lectures on Fermat's last theorem", (1979).
- D.J.Newman, and M.Slater, "Waring's problem for the ring of polynomials", J.Number Theory 11, 477-487 (1979).
- W.Hayman, "Research problems in function theory", book, 1965.
- H. Cartan, "Sur les zeros des combinaisons lineaires de p fonctions holomrphes donnes, book, Mathematica Cluj 7 (1933), 5--29.