Tseitin's tautologies and lower bounds for Nullstellensatz proofs

We use the known linear lower bound for Tseitin's tautologies for establishing linear lower bounds on the degree of Nullstellensatz proofs (in the usual boolean setting) for explicitly constructed systems of polynomials of a constant (in our construction 6) degree. It holds over any field of characteristic distinct from 2. Previously, a linear lower bound was proved for an explicitly constructed system of polynomials of a logarithmic degree.

[1]  Jan Krajícek,et al.  Lower bounds on Hilbert's Nullstellensatz and propositional proofs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[2]  Alasdair Urquhart,et al.  Formal Languages]: Mathematical Logic--mechanical theorem proving , 2022 .

[3]  Paul Beame,et al.  More on the relative strength of counting principles , 1996, Proof Complexity and Feasible Arithmetics.

[4]  A RazborovAlexander Lower bounds for the polynomial calculus , 1998 .

[5]  Zvi Galil,et al.  On the Complexity of Regular Resolution and the Davis-Putnam Procedure , 1977, Theor. Comput. Sci..

[6]  Alasdair Urquhart,et al.  The Complexity of Propositional Proofs , 1995, Bulletin of Symbolic Logic.

[7]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[8]  Noaï Fitchas,et al.  Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen‐Suslin) pour le Calcul Formel , 1990 .

[9]  W. Brownawell Bounds for the degrees in the Nullstellensatz , 1987 .

[10]  Alexander A. Razborov,et al.  Lower bounds for the polynomial calculus , 1998, computational complexity.

[11]  Russell Impagliazzo,et al.  The relative complexity of NP search problems , 1995, STOC '95.

[12]  Russell Impagliazzo,et al.  Lower bounds for the polynomial calculus and the Gröbner basis algorithm , 1999, computational complexity.

[13]  Noga Alon,et al.  Eigenvalues and expanders , 1986, Comb..

[14]  Michael Eugene Stillman,et al.  On the Complexity of Computing Syzygies , 1988, J. Symb. Comput..

[15]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[16]  A. Seidenberg Constructions in algebra , 1974 .

[17]  M. Murty Ramanujan Graphs , 1965 .