A class of mechanically decidable problems beyond Tarski’s model

Abstract

By means of dimension-decreasing method and cell-decomposition, a practical algorithm is proposed to decide the positivity of a certain class of symmetric polynomials, the numbers of whose elements are variable. This is a class of mechanically decidable problems beyond Tarski model. To implement the algorithm, a program nprove written in maple is developed which can decide the positivity of these polynomials rapidly.

Mechanical decision for a class of integral inequalities

“…A complete theoretical result is given. Similar to [1,37], the definitions and notations are as follows.…”

“…[1] did the same thing by using an inequality proving package BOTTEMA. A recent paper [37], published in Science in China, also gave a practical algorithm to decide the nonnegativity of a class of polynomials with uncertain number of elements. A program "nprove" implemented in Maple platform can do this efficiently [1,37].…”

“…In [37], a class of symmetric polynomial inequalities with degree less than 5 is successfully transformed into finitely many inequalities independent of n by using Timfote dimension-decreasing algorithm [7][8][9]. The nonnegativity of these polynomials can be decided by using the inequality-proving package BOTTEMA [1,[10][11][12].…”

“…The nonnegativity of these polynomials can be decided by using the inequality-proving package BOTTEMA [1,[10][11][12]. How to generalize the result of [37] to deal with the polynomials with degree greater than 5 or coefficients depending on n? This is a challenging problem.…”

“…In this paper, a class of integral inequalities is transformed into homogeneous symmetric polynomial inequalities beyond Tarski model, where the number of elements of the polynomial, say n, is also a variable and the coefficients are functions of n. This is closely associated with some open problems formulated recently by Yang et al [37]. Using Timofte's dimension-decreasing method for symmetric polynomial inequalities [7][8][9], combined with the inequality-proving package BOTTEMA [1,[10][11][12] and a program of implementing successive difference substitution [1,[13][14][15][16][17], we provide a procedure to decide the nonnegativity of the corresponding polynomial inequality, hence the original integral inequality is mechanically decidable.…”