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I think Combinatorial Nullstellensatz, and all its applications that I am aware of, fall under category A as well. Here, we construct a polynomial that is explicit enough so that we can determine the coefficient of its leading monomial. The answer by Gjergji Zaimi covers this approach nicely. If we want estimates on the total number of non-vanishing points, instead of just existence, then we can sometimes use the result by Alon-Füredi, as it was recently done by Clark, Forrow and Schmitt in this paper.

I think Combinatorial Nullstellensatz, and all its applications that I am aware of, fall under category A as well. Here, we construct a polynomial that is explicit enough so that we can determine the coefficient of its leading monomial. The answer by Gjergji Zaimi covers this approach nicely. If we want estimates on the total number of non-vanishing points, instead of just existence, then we can sometimes use the result by Alon-Füredi, as it was recently done by Clark, Forrow and Schmitt in this paper.

• Aart Blokhuis, Polynomials in finite geometries and combinatorics.
• A. A. Bruen, J. C. Fisher, The Jamison method in galois geometries.
• Noga Alon, Discrete mathematics: methods and challenges.
• Gyula Károlyi, The polynomial method in additive combinatorics.
• Simeon Ball, Polynomials in Finite Geometries and The Polynomial Method in Galois Geometries.
• Terence Tao and Van Vu, Chapter 9 in Additive Combinatorics.
• Zeev Dvir, Incidence Theorems and Their Applications.
• Terence Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory.
• Peter Sziklai, Polynomials in finite geometries and Applications of Polynomials over Finite Fields.

• Aart Blokhuis, Polynomials in finite geometries and combinatorics.
• A. A. Bruen, J. C. Fisher, The Jamison method in galois geometries.
• Noga Alon, Discrete mathematics: methods and challenges.
• Gyula Károlyi, The polynomial method in additive combinatorics.
• Simeon Ball, Polynomials in Finite Geometries and The Polynomial Method in Galois Geometries.
• Terence Tao and Van Vu, Chapter 9 in Additive Combinatorics.
• Zeev Dvir, Incidence Theorems and Their Applications.
• Terence Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory.
• Peter Sziklai, Polynomials in finite geometries and Applications of Polynomials over Finite Fields.
• Larry Guth, Polynomial Method in Combinatorics.

• Aart Blokhuis, Polynomials in finite geometries and combinatorics.
• A. A. Bruen, J. C. Fisher, The Jamison method in galois geometries.
• Noga Alon, Discrete mathematics: methods and challenges.
• Gyula Károlyi, The polynomial method in additive combinatorics.
• Simeon Ball, Polynomials in Finite Geometries and The Polynomial Method in Galois Geometries.
• Terence Tao and Van Vu, Chapter 9 in Additive Combinatorics.
• Zeev Dvir, Incidence Theorems and Their Applications.
• Terence Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory.

• Aart Blokhuis, Polynomials in finite geometries and combinatorics.
• A. A. Bruen, J. C. Fisher, The Jamison method in galois geometries.
• Noga Alon, Discrete mathematics: methods and challenges.
• Gyula Károlyi, The polynomial method in additive combinatorics.
• Simeon Ball, Polynomials in Finite Geometries and The Polynomial Method in Galois Geometries.
• Terence Tao and Van Vu, Chapter 9 in Additive Combinatorics.
• Zeev Dvir, Incidence Theorems and Their Applications.
• Terence Tao, Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory.
• Peter Sziklai, Polynomials in finite geometries and Applications of Polynomials over Finite Fields.