Homotopies Exploiting Newton Polytopes for Solving Sparse Polynomial Systems

1994 ◽  
Vol 31 (3) ◽  
pp. 915-930 ◽  
Author(s):  
Jan Verschelde ◽  
Pierre Verlinden ◽  
Ronald Cools
Keyword(s):  
Polynomial Systems ◽  
Newton Polytopes ◽  
Sparse Polynomial ◽  
Sparse Polynomial Systems

scholarly journals Criteria for strict monotonicity of the mixed volume of convex polytopes

2019 ◽  
Vol 19 (4) ◽  
pp. 527-540 ◽  
Author(s):  
Frédéric Bihan ◽  
Ivan Soprunov
Keyword(s):  
Convex Hull ◽  
Convex Polytopes ◽  
Mixed Volume ◽  
Polynomial Systems ◽  
Strict Monotonicity ◽  
Newton Polytopes ◽  
Sparse Polynomial ◽  
Geometric Result ◽  
Cramer’S Rule ◽  
Sparse Polynomial Systems

Abstract Let P1, …, Pn and Q1, …, Qn be convex polytopes in ℝn with Pi ⊆ Qi. It is well-known that the mixed volume is monotone: V(P1, …, Pn) ≤ V(Q1, …, Qn). We give two criteria for when this inequality is strict in terms of essential collections of faces as well as mixed polyhedral subdivisions. This geometric result allows us to characterize sparse polynomial systems with Newton polytopes P1, …, Pn whose number of isolated solutions equals the normalized volume of the convex hull of P1 ∪ … ∪ Pn. In addition, we obtain an analog of Cramer’s rule for sparse polynomial systems.

scholarly journals Affine solution sets of sparse polynomial systems

2013 ◽  
Vol 51 ◽  
pp. 34-54 ◽  
Author(s):  
María Isabel Herrero ◽  
Gabriela Jeronimo ◽  
Juan Sabia
Keyword(s):  
Polynomial Systems ◽  
Solution Sets ◽  
Sparse Polynomial ◽  
Sparse Polynomial Systems

scholarly journals Decomposable sparse polynomial systems

2021 ◽  
Vol 11 (1) ◽  
pp. 53-59
Author(s):  
Taylor Brysiewicz ◽  
Jose Israel Rodriguez ◽  
Frank Sottile ◽  
Thomas Yahl
Keyword(s):  
Polynomial Systems ◽  
Sparse Polynomial ◽  
Sparse Polynomial Systems
Sign in / Sign up

Export Citation Format

Share Document