Counting isolated points outside the image of a polynomial map.

In the map (1.4), the set HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi class="MJX-tex-caligraphic" mathvariant="script">I</mi><mi>f</mi><mi mathvariant="normal"> </mi></msubsup></mstyle></mtd></mtr></mtable></math> ht C SP sp \ K results from intersections of vertical lines with the line { I y i SB 1 sb = I y i SB 2 sb }. Then, the polynomial I g i SB 2 sb is written as Graph HT <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><munderover><mo> </mo><mi>j</mi><mo>=</mo><mn>0</mn><msub><mi>m</mi><mi>i</mi></msub></munderover><msup><mi>t</mi><mi>j</mi></msup><msub><mi>g</mi><mn>2</mn><mo>,</mo><mi>j</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>-</mo><msup><mi>s</mi><mi>v</mi></msup><msup><mi>t</mi><mi>w</mi></msup><msub><mi>y</mi><mn>2</mn></msub><mo>,</mo></mstyle></mtd></mtr></mtable></math> ht where ( I v i , I w i ) N SP 2 sp . This shows that for each above I q i , there exists a unique subset (5.13) identifying a subset of HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign=" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mstyle displaystyle="true"><msubsup><mi>m</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><msub><mi>m</mi><mi>i</mi></msub><mo>+</mo><msubsup><mi>n</mi><mi>i</mi><mn>2</mn></msubsup></mstyle></mtd></mtr></mtable></math> ht - I n SB i sb i + I m SB i sb n SB i sb i isolated points in V SP sp ( I G i ) V SP sp ( I H i ) V SP sp ( I K i ). [Extracted from the article]