- 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.